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Theorem ax10 32700
 Description: Rederivation of axiom ax-10 1965 from ax-c7 32690 and other older axioms. See axc7 1992 for the derivation of ax-c7 32690 from ax-10 1965. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax10 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Proof of Theorem ax10
StepHypRef Expression
1 ax-c4 32689 . . 3 (∀𝑥(∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
2 ax-c5 32688 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝑥𝜑)
3 ax-c4 32689 . . . . 5 (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))
4 id 22 . . . . 5 (∀𝑥𝜑 → ∀𝑥𝜑)
53, 4mpg 1700 . . . 4 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
62, 5nsyl 127 . . 3 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑)
71, 6mpg 1700 . 2 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
8 ax-c7 32690 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥𝜑)
97, 8nsyl4 151 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1466 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-c5 32688  ax-c4 32689  ax-c7 32690 This theorem is referenced by:  hba1-o  32702  axc5c711  32722  equidq  32728
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