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Theorem id1 60
Description: Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This version is proved directly from the axioms for demonstration purposes. This proof is a very popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 16 (PDF p. 22) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. For a shorter version of the proof that takes advantage of previously proved theorems, see id 59.
Assertion
Ref Expression
id1 |- (ph -> ph)
Proof of Theorem id1
StepHypRef Expression
1 ax-1 4 . 2 |- (ph -> (ph -> ph))
2 ax-1 4 . . 3 |- (ph -> ((ph -> ph) -> ph))
3 ax-2 5 . . 3 |- ((ph -> ((ph -> ph) -> ph)) -> ((ph -> (ph -> ph)) -> (ph -> ph)))
42, 3ax-mp 7 . 2 |- ((ph -> (ph -> ph)) -> (ph -> ph))
51, 4ax-mp 7 1 |- (ph -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-mp 7
Copyright terms: Public domain