Theorem axc711to11

Description: Rederivation of ax-11 from axc711 . Note that ax-c7 and ax-11 are not used by the rederivation. (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc711to11	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		axc711toc7	⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
2	1	con4i	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
3		axc711	⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 )
4	3	alimi	⊢ ( ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
5	2 4	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )