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Theorem bj-equsal2 34799
Description: One direction of equsal 2040. (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 34797 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  ph )
3 bj-equsal2.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43a2i 13 . . 3  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) )
54alimi 1638 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ps )
)
62, 5sylbir 213 1  |-  ( ph  ->  A. x ( x  =  y  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396   F/wnf 1621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by:  bj-equsal  34800
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