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Theorem id1 20
 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 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. 17 (PDF p. 23) 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 19. (Contributed by NM, 5-Aug-1993.) Use id 19 instead. (New usage is discouraged.)
Assertion
Ref Expression
id1 (φφ)

Proof of Theorem id1
StepHypRef Expression
1 ax-1 5 . 2 (φ → (φφ))
2 ax-1 5 . . 3 (φ → ((φφ) → φ))
3 ax-2 6 . . 3 ((φ → ((φφ) → φ)) → ((φ → (φφ)) → (φφ)))
42, 3ax-mp 7 . 2 ((φ → (φφ)) → (φφ))
51, 4ax-mp 7 1 (φφ)
 Colors of variables: wff set class Syntax hints:   → wi 4 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7 This theorem is referenced by: (None)
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