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Theorem bj-axc10v 31271
Description: Version of axc10 2059 with a dv condition, which does not require ax-13 2054. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc10v  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-axc10v
StepHypRef Expression
1 ax6v 1796 . . 3  |-  -.  A. x  -.  x  =  y
2 con3 140 . . . 4  |-  ( ( x  =  y  ->  A. x ph )  -> 
( -.  A. x ph  ->  -.  x  =  y ) )
32al2imi 1684 . . 3  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ( A. x  -.  A. x ph  ->  A. x  -.  x  =  y ) )
41, 3mtoi 182 . 2  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  -.  A. x  -.  A. x ph )
5 axc7 1913 . 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 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-12 1906
This theorem depends on definitions:  df-bi 189  df-ex 1661
This theorem is referenced by: (None)
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