Theorem axc7 2310

Description: Show that the original axiom ax-c7 37743 can be derived from ax-10 2137 (hbn1 2138), sp 2176 and propositional calculus. See ax10fromc7 37753 for the rederivation of ax-10 2137 from ax-c7 37743.

Normally, axc7 2310 should be used rather than ax-c7 37743, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Assertion

Ref	Expression
axc7	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Proof of Theorem axc7

Step	Hyp	Ref	Expression
1		sp 2176	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2138	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1782

This theorem is referenced by:  modal-b  2312  axc10  2384  hbntg  34765  bj-modalb  35582  bj-axc10v  35659  axc5c4c711  43145  hbntal  43299