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Theorem axc7 2128
Description: Show that the original axiom ax-c7 33647 can be derived from ax-10 2016 (hbn1 2017) , sp 2051 and propositional calculus. See ax10fromc7 33657 for the rederivation of ax-10 2016 from ax-c7 33647.

Normally, axc7 2128 should be used rather than ax-c7 33647, 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 2051 . 2 (∀𝑥𝜑𝜑)
2 hbn1 2017 . 2 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
31, 2nsyl4 156 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by:  modal-b  2138  axc10  2251  hbntg  31409  bj-modalb  32345  bj-axc10v  32356  axc5c4c711  38081  hbntal  38248
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