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Theorem axc7 1911
Description: Show that the original axiom ax-c7 32210 can be derived from ax-10 1886 and others. See ax10 32220 for the rederivation of ax-10 1886 from ax-c7 32210.

Normally, axc7 1911 should be used rather than ax-c7 32210, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Assertion
Ref Expression
axc7  |-  ( -. 
A. x  -.  A. x ph  ->  ph )

Proof of Theorem axc7
StepHypRef Expression
1 sp 1909 . 2  |-  ( A. x ph  ->  ph )
2 hbn1 1887 . 2  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
31, 2nsyl4 147 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  axc7e  1912  modal-b  1913  axc10  2057  hbntg  30280  bj-axc10v  31098  axc5c4c711  36437  hbntal  36605
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