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Theorem axc4 1910
Description: Show that the original axiom ax-c4 32165 can be derived from ax-4 1678 and others. See ax4 32175 for the rederivation of ax-4 1678 from ax-c4 32165.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Assertion
Ref Expression
axc4  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)

Proof of Theorem axc4
StepHypRef Expression
1 sp 1909 . . . 4  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 123 . . 3  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 hbn1 1887 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 hbn1 1887 . . . . 5  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 132 . . . 4  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1680 . . 3  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 18 . 2  |-  ( A. x ph  ->  A. x A. x ph )
8 alim 1679 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x A. x ph  ->  A. x ps ) )
97, 8syl5 33 1  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
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:  axc5c4c711  36393
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