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Theorem ax10fromc7 34696
Description: Rederivation of axiom ax-10 2174 from ax-c7 34686, ax-c4 34685, ax-c5 34684, ax-gen 1870 and propositional calculus. See axc7 2296 for the derivation of ax-c7 34686 from ax-10 2174. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax10fromc7 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Proof of Theorem ax10fromc7
StepHypRef Expression
1 ax-c4 34685 . . 3 (∀𝑥(∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
2 ax-c5 34684 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝑥𝜑)
3 ax-c4 34685 . . . . 5 (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))
4 id 22 . . . . 5 (∀𝑥𝜑 → ∀𝑥𝜑)
53, 4mpg 1872 . . . 4 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
62, 5nsyl 137 . . 3 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ¬ ∀𝑥𝜑)
71, 6mpg 1872 . 2 (∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
8 ax-c7 34686 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝑥𝜑 → ∀𝑥𝜑)
97, 8nsyl4 157 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-c5 34684  ax-c4 34685  ax-c7 34686
This theorem is referenced by:  hba1-o  34698  axc5c711  34719  equidq  34725
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