Theorem axc7 2335

Description: Show that the original axiom ax-c7 36025 can be derived from ax-10 2144 (hbn1 2145) , sp 2181 and propositional calculus. See ax10fromc7 36035 for the rederivation of ax-10 2144 from ax-c7 36025.

Normally, axc7 2335 should be used rather than ax-c7 36025, 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 2181	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2145	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 161	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by:  modal-b  2337  axc10  2402  hbntg  33054  bj-modalb  34054  bj-axc10v  34119  axc5c4c711  40739  hbntal  40893