Theorem equid1ALT 2176

Description: Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2158 from older axioms ax-6o 2137 and ax-9o 2138. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equid1ALT	⊢ x = x

Proof of Theorem equid1ALT

Step	Hyp	Ref	Expression
1		ax-12o 2142	. . . . 5 ⊢ (¬ ∀x x = x → (¬ ∀x x = x → (x = x → ∀x x = x)))
2	1	pm2.43i 43	. . . 4 ⊢ (¬ ∀x x = x → (x = x → ∀x x = x))
3	2	alimi 1559	. . 3 ⊢ (∀x ¬ ∀x x = x → ∀x(x = x → ∀x x = x))
4		ax-9o 2138	. . 3 ⊢ (∀x(x = x → ∀x x = x) → x = x)
5	3, 4	syl 15	. 2 ⊢ (∀x ¬ ∀x x = x → x = x)
6		ax-6o 2137	. 2 ⊢ (¬ ∀x ¬ ∀x x = x → x = x)
7	5, 6	pm2.61i 156	1 ⊢ x = x

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-6o 2137  ax-9o 2138  ax-12o 2142

This theorem is referenced by: (None)