Theorem axc11nfromc11 39728

Description: Rederivation of ax-c11n 39690 from original version ax-c11 39689. See Theorem axc11 2461 for the derivation of ax-c11 39689 from ax-c11n 39690.

This theorem should not be referenced in any proof. Instead, use ax-c11n 39690 above so that uses of ax-c11n 39690 can be more easily identified, or use aecom-o 39703 when this form is needed for studies involving ax-c11 39689 and omitting ax-5 1939. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11nfromc11	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11nfromc11

Step	Hyp	Ref	Expression
1		ax-c11 39689	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 53	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2046	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1840	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-c11 39689

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by: (None)