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Theorem axsep2 2694
Description: A less restrictive version of the Separation Scheme axsep 2692, where variables x and z can both appear free in the wff ph, which can therefore be thought of as ph(x, z). This version was derived from the more restrictive ax-sep 2693 with no additional set theory axioms.
Assertion
Ref Expression
axsep2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
Distinct variable groups:   x,y,z   ph,y
Proof of Theorem axsep2
StepHypRef Expression
1 a9e 1121 . 2 |- E.w w = z
2 ax-sep 2693 . . . 4 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
3 elequ2 1133 . . . . . . . . . . 11 |- (w = z -> (x e. w <-> x e. z))
43biimprd 154 . . . . . . . . . 10 |- (w = z -> (x e. z -> x e. w))
54pm4.71rd 637 . . . . . . . . 9 |- (w = z -> (x e. z <-> (x e. w /\ x e. z)))
65anbi1d 615 . . . . . . . 8 |- (w = z -> ((x e. z /\ ph) <-> ((x e. w /\ x e. z) /\ ph)))
7 anass 439 . . . . . . . 8 |- (((x e. w /\ x e. z) /\ ph) <-> (x e. w /\ (x e. z /\ ph)))
86, 7syl6bb 534 . . . . . . 7 |- (w = z -> ((x e. z /\ ph) <-> (x e. w /\ (x e. z /\ ph))))
98bibi2d 616 . . . . . 6 |- (w = z -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. w /\ (x e. z /\ ph)))))
109albidv 1273 . . . . 5 |- (w = z -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
1110exbidv 1274 . . . 4 |- (w = z -> (E.yA.x(x e. y <-> (x e. z /\ ph)) <-> E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
122, 11mpbiri 194 . . 3 |- (w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
131219.23aiv 1290 . 2 |- (E.w w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
141, 13ax-mp 7 1 |- E.yA.x(x e. y <-> (x e. z /\ ph))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-8 961  ax-9 962  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-sep 2693
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978
Copyright terms: Public domain