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Theorem axc7 2296
Description: Show that the original axiom ax-c7 34692 can be derived from ax-10 2174 (hbn1 2175) , sp 2207 and propositional calculus. See ax10fromc7 34702 for the rederivation of ax-10 2174 from ax-c7 34692.

Normally, axc7 2296 should be used rather than ax-c7 34692, 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 2207 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2175 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 157 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  modal-b  2307  axc10  2414  hbntg  32046  bj-modalb  33042  bj-axc10v  33053  axc5c4c711  39128  hbntal  39294
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