Theorem axc5c711toc7 39508

Description: Rederivation of ax-c7 39473 from axc5c711 39506. Note that ax-c7 39473 and ax-11 2190 are not used by the rederivation. The use of alimi 1830 (which uses ax-c5 39471) is allowed since we have already proved axc5c711toc5 39507. (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 39485	. . . . . 6 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2	1	con3i 154	. . . . 5 ⊢ (¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑)
3	2	alimi 1830	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4	3	sps-o 39496	. . 3 ⊢ (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5	4	con3i 154	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑)
6		pm2.21 123	. 2 ⊢ (¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑))
7		axc5c711 39506	. 2 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
8	5, 6, 7	3syl 18	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-11 2190  ax-c5 39471  ax-c4 39472  ax-c7 39473

This theorem is referenced by:  axc5c711to11  39509