Theorem axc5c711to11 36672

Description: Rederivation of ax-11 2158 from axc5c711 36669. Note that ax-c7 36636 and ax-11 2158 are not used by the rederivation. The use of alimi 1819 (which uses ax-c5 36634) is allowed since we have already proved axc5c711toc5 36670. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c711to11	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Proof of Theorem axc5c711to11

Step	Hyp	Ref	Expression
1		axc5c711toc7 36671	. . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ¬ ∀𝑥∀𝑦𝜑)
2	1	con4i 114	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
3		pm2.21 123	. . . . . 6 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → (∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑))
4		axc5c711 36669	. . . . . 6 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
5	3, 4	syl 17	. . . . 5 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑)
6	5	alimi 1819	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
7		axc5c711toc7 36671	. . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
8	6, 7	nsyl4 161	. . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
9	8	alimi 1819	. 2 ⊢ (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
10	2, 9	syl 17	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-11 2158  ax-c5 36634  ax-c4 36635  ax-c7 36636

This theorem is referenced by: (None)