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Theorem axc10 2060
Description: Show that the original axiom ax-c10 32167 can be derived from ax6 2059 and others. See ax6fromc10 32177 for the rederivation of ax6 2059 from ax-c10 32167.

Normally, axc10 2060 should be used rather than ax-c10 32167, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Assertion
Ref Expression
axc10  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )

Proof of Theorem axc10
StepHypRef Expression
1 ax6 2059 . . 3  |-  -.  A. x  -.  x  =  y
2 con3 139 . . . 4  |-  ( ( x  =  y  ->  A. x ph )  -> 
( -.  A. x ph  ->  -.  x  =  y ) )
32al2imi 1683 . . 3  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ( A. x  -.  A. x ph  ->  A. x  -.  x  =  y ) )
41, 3mtoi 181 . 2  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  -.  A. x  -.  A. x ph )
5 axc7 1914 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
64, 5syl 17 1  |-  ( A. x ( x  =  y  ->  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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660
This theorem is referenced by: (None)
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