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Theorem axc5c711toc7 34524
Description: Rederivation of ax-c7 34489 from axc5c711 34522. Note that ax-c7 34489 and ax-11 2074 are not used by the rederivation. The use of alimi 1779 (which uses ax-c5 34487) is allowed since we have already proved axc5c711toc5 34523. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c711toc7 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Proof of Theorem axc5c711toc7
StepHypRef Expression
1 hba1-o 34501 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
21con3i 150 . . . . 5 (¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑)
32alimi 1779 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
43sps-o 34512 . . 3 (∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con3i 150 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑)
6 pm2.21 120 . 2 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → (∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥𝜑))
7 axc5c711 34522 . 2 ((∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
85, 6, 73syl 18 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-11 2074  ax-c5 34487  ax-c4 34488  ax-c7 34489
This theorem is referenced by:  axc5c711to11  34525
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