ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  id1 Structured version   GIF version

Theorem id1 18
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 17. (Contributed by NM, 5-Aug-1993.)
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
  Copyright terms: Public domain W3C validator