Theorem axc5c711toc7 34205

Description: Rederivation of ax-c7 34170 from axc5c711 34203. Note that ax-c7 34170 and ax-11 2034 are not used by the rederivation. The use of alimi 1739 (which uses ax-c5 34168) is allowed since we have already proved axc5c711toc5 34204. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c711toc7	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Proof of Theorem axc5c711toc7

Step	Hyp	Ref	Expression
1		hba1-o 34182	. . . . . 6 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2	1	con3i 150	. . . . 5 ⊢ (¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑)
3	2	alimi 1739	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4	3	sps-o 34193	. . 3 ⊢ (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5	4	con3i 150	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑)
6		pm2.21 120	. 2 ⊢ (¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑))
7		axc5c711 34203	. 2 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
8	5, 6, 7	3syl 18	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-11 2034  ax-c5 34168  ax-c4 34169  ax-c7 34170

This theorem is referenced by:  axc5c711to11  34206