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Theorem axc7 2328
Description: Show that the original axiom ax-c7 36180 can be derived from ax-10 2143 (hbn1 2144) , sp 2181 and propositional calculus. See ax10fromc7 36190 for the rederivation of ax-10 2143 from ax-c7 36180.

Normally, axc7 2328 should be used rather than ax-c7 36180, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Assertion
Ref Expression
axc7 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Proof of Theorem axc7
StepHypRef Expression
1 sp 2181 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2144 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 161 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  modal-b  2330  axc10  2395  hbntg  33164  bj-modalb  34164  bj-axc10v  34231  axc5c4c711  41102  hbntal  41256
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