Theorem axc7 2351

Description: Show that the original axiom ax-c7 40145 can be derived from ax-10 2180 (hbn1 2181), sp 2223 and propositional calculus. See ax10fromc7 40155 for the rederivation of ax-10 2180 from ax-c7 40145.

Normally, axc7 2351 should be used rather than ax-c7 40145, 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 2223	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2181	. 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 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

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

This theorem is used by:  modal-b  2353  axc10  2417  hbntg  36758  bj-modalb  37811  bj-axc10v  37896  axc5c4c711  45588  hbntal  45738