Theorem axc5c711to11 38922

Description: Rederivation of ax-11 2157 from axc5c711 38919. Note that ax-c7 38886 and ax-11 2157 are not used by the rederivation. The use of alimi 1811 (which uses ax-c5 38884) is allowed since we have already proved axc5c711toc5 38920. (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 38921	. . 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 38919	. . . . . 6 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
5	3, 4	syl 17	. . . . 5 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑)
6	5	alimi 1811	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
7		axc5c711toc7 38921	. . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
8	6, 7	nsyl4 158	. . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
9	8	alimi 1811	. 2 ⊢ (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
10	2, 9	syl 17	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-11 2157  ax-c5 38884  ax-c4 38885  ax-c7 38886

This theorem is referenced by: (None)