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Theorem axc7 2309
Description: Show that the original axiom ax-c7 38219 can be derived from ax-10 2136 (hbn1 2137), sp 2175 and propositional calculus. See ax10fromc7 38229 for the rederivation of ax-10 2136 from ax-c7 38219.

Normally, axc7 2309 should be used rather than ax-c7 38219, 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 2175 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2137 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 158 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-ex 1781
This theorem is referenced by:  modal-b  2311  axc10  2383  hbntg  35247  bj-modalb  36058  bj-axc10v  36135  axc5c4c711  43623  hbntal  43777
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