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Theorem axc7 1939
Description: Show that the original axiom ax-c7 32457 can be derived from ax-10 1915 and others. See ax10 32467 for the rederivation of ax-10 1915 from ax-c7 32457.

Normally, axc7 1939 should be used rather than ax-c7 32457, 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 1937 . 2  |-  ( A. x ph  ->  ph )
2 hbn1 1916 . 2  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
31, 2nsyl4 148 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664
This theorem is referenced by:  axc7e  1940  modal-b  1941  axc10  2096  hbntg  30452  bj-axc10v  31318  axc5c4c711  36752  hbntal  36920
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