Theorem axc7 2309

Description: Show that the original axiom ax-c7 38219 can be derived from ax-10 2136 (hbn1 2137), sp 2175 and propositional calculus. See ax10fromc7 38229 for the rederivation of ax-10 2136 from ax-c7 38219.

Normally, axc7 2309 should be used rather than ax-c7 38219, 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 2175	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2137	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

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 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by:  modal-b  2311  axc10  2383  hbntg  35247  bj-modalb  36058  bj-axc10v  36135  axc5c4c711  43623  hbntal  43777