Theorem axc7 2292

Description: Show that the original axiom ax-c7 35044 can be derived from ax-10 2135 (hbn1 2136) , sp 2167 and propositional calculus. See ax10fromc7 35054 for the rederivation of ax-10 2135 from ax-c7 35044.

Normally, axc7 2292 should be used rather than ax-c7 35044, 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 2167	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2136	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-ex 1824

This theorem is referenced by:  modal-b  2294  axc10  2349  hbntg  32303  bj-modalb  33298  bj-axc10v  33309  axc5c4c711  39567  hbntal  39723