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Theorem axc7 1992
Description: Show that the original axiom ax-c7 32690 can be derived from ax-10 1965 and others. See ax10 32700 for the rederivation of ax-10 1965 from ax-c7 32690.

Normally, axc7 1992 should be used rather than ax-c7 32690, 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 1990 . 2 (∀𝑥𝜑𝜑)
2 hbn1 1966 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 151 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-ex 1693
This theorem is referenced by:  axc7e  1993  modal-b  1994  axc10  2143  hbntg  30603  bj-modalb  31494  bj-axc10v  31505  axc5c4c711  37109  hbntal  37276
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