Theorem axc11nfromc11

Description: Rederivation of ax-c11n from original version ax-c11 . See theorem axc11 for the derivation of ax-c11 from ax-c11n .

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

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

Proof

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