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Theorem axc7 2292
Description: Show that the original axiom ax-c7 35044 can be derived from ax-10 2135 (hbn1 2136) , sp 2167 and propositional calculus. See ax10fromc7 35054 for the rederivation of ax-10 2135 from ax-c7 35044.

Normally, axc7 2292 should be used rather than ax-c7 35044, 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 2167 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2136 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 158 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163
This theorem depends on definitions:  df-bi 199  df-ex 1824
This theorem is referenced by:  modal-b  2294  axc10  2349  hbntg  32303  bj-modalb  33298  bj-axc10v  33309  axc5c4c711  39567  hbntal  39723
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