Theorem axc5c711to11

Description: Rederivation of ax-11 from axc5c711 . Note that ax-c7 and ax-11 are not used by the rederivation. The use of alimi (which uses ax-c5 ) is allowed since we have already proved axc5c711toc5 . (Contributed by NM, 19-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc5c711to11	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		axc5c711toc7	⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
2	1	con4i	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
3		pm2.21	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )
4		axc5c711	⊢ ( ( ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 )
5	3 4	syl	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 )
6	5	alimi	⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 )
7		axc5c711toc7	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
8	6 7	nsyl4	⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 )
9	8	alimi	⊢ ( ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
10	2 9	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )