Theorem axc7 2320

Description: Show that the original axiom ax-c7 39004 can be derived from ax-10 2146 (hbn1 2147), sp 2188 and propositional calculus. See ax10fromc7 39014 for the rederivation of ax-10 2146 from ax-c7 39004.

Normally, axc7 2320 should be used rather than ax-c7 39004, 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 2188	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2147	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  modal-b  2322  axc10  2387  hbntg  35868  bj-modalb  36781  bj-axc10v  36858  axc5c4c711  44518  hbntal  44670