Theorem axc7 2348

Description: Show that the original axiom ax-c7 39473 can be derived from ax-10 2174 (hbn1 2175), sp 2217 and propositional calculus. See ax10fromc7 39483 for the rederivation of ax-10 2174 from ax-c7 39473.

Normally, axc7 2348 should be used rather than ax-c7 39473, 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 2217	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2175	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  modal-b  2350  axc10  2415  hbntg  36117  bj-modalb  37157  bj-axc10v  37242  axc5c4c711  44941  hbntal  45093