Theorem axc7 2117

Description: Show that the original axiom ax-c7 33188 can be derived from ax-10 2006 (hbn1 2007) , sp 2041 and propositional calculus. See ax10fromc7 33198 for the rederivation of ax-10 2006 from ax-c7 33188.

Normally, axc7 2117 should be used rather than ax-c7 33188, 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 2041	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2007	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 155	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  modal-b  2127  axc10  2240  hbntg  30955  bj-modalb  31893  bj-axc10v  31904  axc5c4c711  37624  hbntal  37790