Theorem axc7 2353

Description: Show that the original axiom ax-c7 39700 can be derived from ax-10 2179 (hbn1 2180), sp 2222 and propositional calculus. See ax10fromc7 39710 for the rederivation of ax-10 2179 from ax-c7 39700.

Normally, axc7 2353 should be used rather than ax-c7 39700, 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 2222	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2180	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 159	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  modal-b  2355  axc10  2420  hbntg  36316  bj-modalb  37384  bj-axc10v  37469  axc5c4c711  45152  hbntal  45303