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Theorem axc7 2271
Description: Show that the original axiom ax-c7 34666 can be derived from ax-10 2160 (hbn1 2161) , sp 2192 and propositional calculus. See ax10fromc7 34676 for the rederivation of ax-10 2160 from ax-c7 34666.

Normally, axc7 2271 should be used rather than ax-c7 34666, 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 2192 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2161 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 156 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-12 2188
This theorem depends on definitions:  df-bi 197  df-ex 1846
This theorem is referenced by:  modal-b  2281  axc10  2389  hbntg  32008  bj-modalb  33004  bj-axc10v  33015  axc5c4c711  39096  hbntal  39263
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