Theorem axc7 2132

Description: Show that the original axiom ax-c7 34170 can be derived from ax-10 2019 (hbn1 2020) , sp 2053 and propositional calculus. See ax10fromc7 34180 for the rederivation of ax-10 2019 from ax-c7 34170.

Normally, axc7 2132 should be used rather than ax-c7 34170, 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 2053	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		hbn1 2020	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	1, 2	nsyl4 156	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  modal-b  2142  axc10  2252  hbntg  31711  bj-modalb  32706  bj-axc10v  32717  axc5c4c711  38602  hbntal  38769