Theorem axc11nfromc11 40026

Description: Rederivation of ax-c11n 39988 from original version ax-c11 39987. See Theorem axc11 2459 for the derivation of ax-c11 39987 from ax-c11n 39988.

This theorem should not be referenced in any proof. Instead, use ax-c11n 39988 above so that uses of ax-c11n 39988 can be more easily identified, or use aecom-o 40001 when this form is needed for studies involving ax-c11 39987 and omitting ax-5 1942. (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 39987	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 53	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2049	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1843	. 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 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-c11 39987

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by: (None)