Theorem axc7 2311

Description: Show that the original axiom ax-c7 37755 can be derived from ax-10 2138 (hbn1 2139), sp 2177 and propositional calculus. See ax10fromc7 37765 for the rederivation of ax-10 2138 from ax-c7 37755.

Normally, axc7 2311 should be used rather than ax-c7 37755, 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 2177	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2139	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  modal-b  2313  axc10  2385  hbntg  34777  bj-modalb  35594  bj-axc10v  35671  axc5c4c711  43160  hbntal  43314