Theorem axc7 2328

Description: Show that the original axiom ax-c7 39390 can be derived from ax-10 2154 (hbn1 2155), sp 2197 and propositional calculus. See ax10fromc7 39400 for the rederivation of ax-10 2154 from ax-c7 39390.

Normally, axc7 2328 should be used rather than ax-c7 39390, 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 2197	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2155	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-ex 1788

This theorem is referenced by:  modal-b  2330  axc10  2395  hbntg  36044  bj-modalb  37074  bj-axc10v  37159  axc5c4c711  44858  hbntal  45010