Theorem axc711 38386

Description: Proof of a single axiom that can replace both ax-c7 38357 and ax-11 2147. See axc711toc7 38388 and axc711to11 38389 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc711	⊢ (¬ ∀𝑥 ¬ ∀𝑦∀𝑥𝜑 → ∀𝑦𝜑)

Proof of Theorem axc711

Step	Hyp	Ref	Expression
1		ax-11 2147	. . . . 5 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
2	1	con3i 154	. . . 4 ⊢ (¬ ∀𝑥∀𝑦𝜑 → ¬ ∀𝑦∀𝑥𝜑)
3	2	alimi 1806	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑦∀𝑥𝜑)
4	3	con3i 154	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑦∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5		ax-c7 38357	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
6	4, 5	syl 17	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑦∀𝑥𝜑 → ∀𝑦𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-11 2147  ax-c7 38357

This theorem is referenced by:  axc711toc7  38388  axc711to11  38389