Theorem idALT 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.) (Proof modification is discouraged.) Use id 19 instead. (New usage is discouraged.)

Assertion

Ref	Expression
idALT	⊢ (𝜑 → 𝜑)

Proof of Theorem idALT

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝜑 → (𝜑 → 𝜑))
2		ax-1 6	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜑) → 𝜑))
3		ax-2 7	. . 3 ⊢ ((𝜑 → ((𝜑 → 𝜑) → 𝜑)) → ((𝜑 → (𝜑 → 𝜑)) → (𝜑 → 𝜑)))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜑 → 𝜑)) → (𝜑 → 𝜑))
5	1, 4	ax-mp 5	1 ⊢ (𝜑 → 𝜑)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by: (None)