Theorem axc7 2314

Description: Show that the original axiom ax-c7 36878 can be derived from ax-10 2140 (hbn1 2141) , sp 2179 and propositional calculus. See ax10fromc7 36888 for the rederivation of ax-10 2140 from ax-c7 36878.

Normally, axc7 2314 should be used rather than ax-c7 36878, 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 2179	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2141	. 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 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-ex 1786

This theorem is referenced by:  modal-b  2316  axc10  2386  hbntg  33760  bj-modalb  34877  bj-axc10v  34954  axc5c4c711  41972  hbntal  42126