Theorem axc7 2352

Description: Show that the original axiom ax-c7 39521 can be derived from ax-10 2178 (hbn1 2179), sp 2221 and propositional calculus. See ax10fromc7 39531 for the rederivation of ax-10 2178 from ax-c7 39521.

Normally, axc7 2352 should be used rather than ax-c7 39521, 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 2221	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2179	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 159	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-ex 1803

This theorem is referenced by:  modal-b  2354  axc10  2419  hbntg  36166  bj-modalb  37205  bj-axc10v  37290  axc5c4c711  44975  hbntal  45127