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Statement List for Metamath Proof Explorer - 601-700 - Page 7 of 107
TypeLabelDescription
Statement
 
Theoremjctird 601 Deduction conjoining a theorem to right of consequent in an implication.
|- (ph -> (ps -> ch))   &   |- (ph -> th)   =>   |- (ph -> (ps -> (ch /\ th)))
 
Theorempm3.43 602 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113.
|- (((ph -> ps) /\ (ph -> ch)) -> (ph -> (ps /\ ch)))
 
Theoremandi 603 Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118.
|- ((ph /\ (ps \/ ch)) <-> ((ph /\ ps) \/ (ph /\ ch)))
 
Theoremandir 604 Distributive law for conjunction.
|- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))
 
Theoremorddi 605 Double distributive law for disjunction.
|- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
 
Theoremanddi 606 Double distributive law for conjunction.
|- (((ph \/ ps) /\ (ch \/ th)) <-> (((ph /\ ch) \/ (ph /\ th)) \/ ((ps /\ ch) \/ (ps /\ th))))
 
Theorembibi2i 607 Inference adding a biconditional to the left in an equivalence.
|- (ph <-> ps)   =>   |- ((ch <-> ph) <-> (ch <-> ps))
 
Theorembibi1i 608 Inference adding a biconditional to the right in an equivalence.
|- (ph <-> ps)   =>   |- ((ph <-> ch) <-> (ps <-> ch))
 
Theorembibi12i 609 The equivalence of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph <-> ch) <-> (ps <-> th))
 
Theoremnegbid 610 Deduction negating both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (-. ps <-> -. ch))
 
Theoremimbi2d 611 Deduction adding an antecedent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th -> ps) <-> (th -> ch)))
 
Theoremimbi1d 612 Deduction adding a consequent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps -> th) <-> (ch -> th)))
 
Theoremorbi2d 613 Deduction adding a left disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th \/ ps) <-> (th \/ ch)))
 
Theoremorbi1d 614 Deduction adding a right disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ th)))
 
Theoremanbi2d 615 Deduction adding a left conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps) <-> (th /\ ch)))
 
Theoremanbi1d 616 Deduction adding a right conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ th)))
 
Theorembibi2d 617 Deduction adding a biconditional to the left in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th <-> ps) <-> (th <-> ch)))
 
Theorembibi1d 618 Deduction adding a biconditional to the right in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> th)))
 
Theoremorbi1 619 Theorem *4.37 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph \/ ch) <-> (ps \/ ch)))
 
Theoremanbi1 620 Theorem *4.36 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph /\ ch) <-> (ps /\ ch)))
 
Theorempm4.22 621 Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ps <-> ch)) -> (ph <-> ch))
 
Theoremimbi1 622 Theorem *4.84 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph -> ch) <-> (ps -> ch)))
 
Theoremimbi2 623 Theorem *4.85 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ch -> ph) <-> (ch -> ps)))
 
Theorembibi1 624 Theorem *4.86 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph <-> ch) <-> (ps <-> ch)))
 
Theoremimbi12d 625 Deduction joining two equivalences to form equivalence of implications.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps -> th) <-> (ch -> ta)))
 
Theoremorbi12d 626 Deduction joining two equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ ta)))
 
Theoremanbi12d 627 Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ ta)))
 
Theorembibi12d 628 Deduction joining two equivalences to form equivalence of biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> ta)))
 
Theorempm4.39 629 Theorem *4.39 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph \/ ps) <-> (ch \/ th)))
 
Theorempm4.38 630 Theorem *4.38 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph /\ ps) <-> (ch /\ th)))
 
Theorembi2anan9 631 Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps /\ ta) <-> (ch /\ et)))
 
Theorembi2anan9r 632 Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((th /\ ph) -> ((ps /\ ta) <-> (ch /\ et)))
 
Theorembi2bian9 633 Deduction joining two biconditionals with different antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps <-> ta) <-> (ch <-> et)))
 
Theorempm4.71 634 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (ph <-> (ph /\ ps)))
 
Theorempm4.71r 635 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed).
|- ((ph -> ps) <-> (ph <-> (ps /\ ph)))
 
Theorempm4.71i 636 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120.
|- (ph -> ps)   =>   |- (ph <-> (ph /\ ps))
 
Theorempm4.71ri 637 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed).
|- (ph -> ps)   =>   |- (ph <-> (ps /\ ph))
 
Theorempm4.71rd 638 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps <-> (ch /\ ps)))
 
Theorempm4.45 639 Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> (ph /\ (ph \/ ps)))
 
Theorempm4.72 640 Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121.
|- ((ph -> ps) <-> (ps <-> (ph \/ ps)))
 
Theoremiba 641 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121.
|- (ph -> (ps <-> (ps /\ ph)))
 
Theoremibar 642 Introduction of antecedent as conjunct.
|- (ph -> (ps <-> (ph /\ ps)))
 
Theorempm5.32 643 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))
 
Theorempm5.32i 644 Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) <-> (ph /\ ch))
 
Theorempm5.32ri 645 Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) <-> (ch /\ ph))
 
Theorempm5.32d 646 Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
 
Theorempm5.32rd 647 Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ch /\ ps) <-> (th /\ ps)))
 
Theorempm5.32da 648 Distribution of implication over biconditional (deduction rule).
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
 
Theorempm5.33 649 Theorem *5.33 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ps -> ch)) <-> (ph /\ ((ph /\ ps) -> ch)))
 
Theorempm5.36 650 Theorem *5.36 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ph <-> ps)) <-> (ps /\ (ph <-> ps)))
 
Theorempm5.42 651 Theorem *5.42 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps -> ch)) <-> (ph -> (ps -> (ph /\ ch))))
 
Theorembianabs 652 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007; replaces ompfl2OLD 10392.)
|- (ph -> (ps <-> (ph /\ ch)))   =>   |- (ph -> (ps <-> ch))
 
Theoremoibabs 653 Absorption of disjunction into equivalence.
|- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
 
Theoremexmid 654 Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic.
|- (ph \/ -. ph)
 
Theorempm2.1 655 Theorem *2.1 of [WhiteheadRussell] p. 101.
|- (-. ph \/ ph)
 
Theorempm2.13 656 Theorem *2.13 of [WhiteheadRussell] p. 101.
|- (ph \/ -. -. -. ph)
 
Theorempm3.24 657 Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction").
|- -. (ph /\ -. ph)
 
Theorempm2.26 658 Theorem *2.26 of [WhiteheadRussell] p. 104.
|- (-. ph \/ ((ph -> ps) -> ps