Theorem axc7 2347

Description: Show that the original axiom ax-c7 40035 can be derived from ax-10 2177 (hbn1 2178), sp 2219 and propositional calculus. See ax10fromc7 40045 for the rederivation of ax-10 2177 from ax-c7 40035.

Normally, axc7 2347 should be used rather than ax-c7 40035, 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 2219	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2178	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 159	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  modal-b  2349  axc10  2414  hbntg  36659  bj-modalb  37712  bj-axc10v  37797  axc5c4c711  45478  hbntal  45629