Theorem axc11nfromc11 36940

Description: Rederivation of ax-c11n 36902 from original version ax-c11 36901. See Theorem axc11 2430 for the derivation of ax-c11 36901 from ax-c11n 36902.

This theorem should not be referenced in any proof. Instead, use ax-c11n 36902 above so that uses of ax-c11n 36902 can be more easily identified, or use aecom-o 36915 when this form is needed for studies involving ax-c11 36901 and omitting ax-5 1913. (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 36901	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2020	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1814	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-c11 36901

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by: (None)