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Theorem axc5c711toc7 33006
Description: Rederivation of ax-c7 32971 from axc5c711 33004. Note that ax-c7 32971 and ax-11 2020 are not used by the rederivation. The use of alimi 1729 (which uses ax-c5 32969) is allowed since we have already proved axc5c711toc5 33005. (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 32983 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
21con3i 148 . . . . 5 (¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑)
32alimi 1729 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
43sps-o 32994 . . 3 (∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con3i 148 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑)
6 pm2.21 118 . 2 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → (∀𝑥𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥𝜑))
7 axc5c711 33004 . 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 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-11 2020  ax-c5 32969  ax-c4 32970  ax-c7 32971
This theorem is referenced by:  axc5c711to11  33007
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