Theorem axc7 2350

Description: Show that the original axiom ax-c7 39637 can be derived from ax-10 2176 (hbn1 2177), sp 2219 and propositional calculus. See ax10fromc7 39647 for the rederivation of ax-10 2176 from ax-c7 39637.

Normally, axc7 2350 should be used rather than ax-c7 39637, 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 2177	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 159	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  modal-b  2352  axc10  2417  hbntg  36273  bj-modalb  37321  bj-axc10v  37406  axc5c4c711  45091  hbntal  45242