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Theorem axc5c7 32997
Description: Proof of a single axiom that can replace ax-c5 32969 and ax-c7 32971. See axc5c7toc5 32998 and axc5c7toc7 32999 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c7 ((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc5c7
StepHypRef Expression
1 ax-c7 32971 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
2 ax-c5 32969 . 2 (∀𝑥𝜑𝜑)
31, 2ja 171 1 ((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c5 32969  ax-c7 32971
This theorem is referenced by:  axc5c7toc5  32998  axc5c7toc7  32999
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