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Theorem axc7 2116
Description: Show that the original axiom ax-c7 32984 can be derived from ax-10 2005 (hbn1 2006) , sp 2040 and propositional calculus. See ax10fromc7 32994 for the rederivation of ax-10 2005 from ax-c7 32984.

Normally, axc7 2116 should be used rather than ax-c7 32984, 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 2040 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2006 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 154 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  modal-b  2126  axc10  2239  hbntg  30761  bj-modalb  31699  bj-axc10v  31710  axc5c4c711  37420  hbntal  37586
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