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Theorem bj-ceqsalgvALT 34858
Description: Alternate proof of bj-ceqsalgv 34857. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalgv.1  |-  F/ x ps
bj-ceqsalgv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bj-ceqsalgvALT  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem bj-ceqsalgvALT
StepHypRef Expression
1 bj-ceqsalgv.1 . 2  |-  F/ x ps
2 bj-ceqsalgv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1623 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 bj-ceqsaltv 34853 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1312 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398   F/wnf 1621    e. wcel 1823
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
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-ex 1618  df-nf 1622  df-clel 2449
This theorem is referenced by: (None)
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