Theorem axc7 2318

Description: Show that the original axiom ax-c7 36585 can be derived from ax-10 2143 (hbn1 2144) , sp 2182 and propositional calculus. See ax10fromc7 36595 for the rederivation of ax-10 2143 from ax-c7 36585.

Normally, axc7 2318 should be used rather than ax-c7 36585, 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 2182	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2144	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 161	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  modal-b  2320  axc10  2384  hbntg  33451  bj-modalb  34584  bj-axc10v  34661  axc5c4c711  41633  hbntal  41787