Theorem equid1ALT 36625

Description: Alternate proof of equid 2022 and equid1 36599 from older axioms ax-c7 36585, ax-c10 36586 and ax-c9 36590. (Contributed by NM, 10-Jan-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-c9 36590	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑥 → (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
2	1	pm2.43i 52	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
3	2	alimi 1819	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
4		ax-c10 36586	. . 3 ⊢ (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥)
5	3, 4	syl 17	. 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥)
6		ax-c7 36585	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥)
7	5, 6	pm2.61i 185	1 ⊢ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-c7 36585  ax-c10 36586  ax-c9 36590

This theorem is referenced by: (None)