Theorem axc7 2317

Description: Show that the original axiom ax-c7 38886 can be derived from ax-10 2141 (hbn1 2142), sp 2183 and propositional calculus. See ax10fromc7 38896 for the rederivation of ax-10 2141 from ax-c7 38886.

Normally, axc7 2317 should be used rather than ax-c7 38886, 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 2183	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2142	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 158	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

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 1967  ax-7 2007  ax-10 2141  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  modal-b  2319  axc10  2390  hbntg  35806  bj-modalb  36717  bj-axc10v  36794  axc5c4c711  44420  hbntal  44573