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Theorem axc4 1809
Description: Show that the original axiom ax-c4 2208 can be derived from ax-4 1612 and others. See ax4 2218 for the rederivation of ax-4 1612 from ax-c4 2208.

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 1808 . . . 4  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 120 . . 3  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 hbn1 1787 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 hbn1 1787 . . . . 5  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 129 . . . 4  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1614 . . 3  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 20 . 2  |-  ( A. x ph  ->  A. x A. x ph )
8 alim 1613 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x A. x ph  ->  A. x ps ) )
97, 8syl5 32 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 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  axc5c4c711  30886
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