Theorem axc711toc7 39941

Description: Rederivation of ax-c7 39910 from axc711 39939. Note that ax-c7 39910 and ax-11 2194 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc711toc7	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Proof of Theorem axc711toc7

Step	Hyp	Ref	Expression
1		hba1-o 39922	. . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2	1	con3i 155	. . . 4 ⊢ (¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑)
3	2	alimi 1844	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4	3	con3i 155	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑)
5		axc711 39939	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑)
6		ax-c5 39908	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
7	4, 5, 6	3syl 19	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-11 2194  ax-c5 39908  ax-c4 39909  ax-c7 39910

This theorem is used by:  axc711to11  39942