Theorem axc7 2316

Description: Show that the original axiom ax-c7 38867 can be derived from ax-10 2139 (hbn1 2140), sp 2181 and propositional calculus. See ax10fromc7 38877 for the rederivation of ax-10 2139 from ax-c7 38867.

Normally, axc7 2316 should be used rather than ax-c7 38867, 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 2181	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2140	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by:  modal-b  2318  axc10  2388  hbntg  35787  bj-modalb  36699  bj-axc10v  36776  axc5c4c711  44397  hbntal  44551