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Theorem bj-equsal2 31338
Description: One direction of equsal 2100. (Contributed by BJ, 30-Sep-2018.)
Hypotheses
Ref Expression
bj-equsal2.1  |-  F/ x ph
bj-equsal2.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
bj-equsal2  |-  ( ph  ->  A. x ( x  =  y  ->  ps ) )

Proof of Theorem bj-equsal2
StepHypRef Expression
1 bj-equsal2.1 . . 3  |-  F/ x ph
21bj-equsal1ti 31336 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  ph )
3 bj-equsal2.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43a2i 14 . . 3  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) )
54alimi 1678 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ps )
)
62, 5sylbir 216 1  |-  ( ph  ->  A. x ( x  =  y  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  bj-equsal  31339
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