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Theorem ceqsalg 3120
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3121. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
ceqsalg.1  |-  F/ x ps
ceqsalg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsalg  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsalg
StepHypRef Expression
1 ceqsalg.1 . 2  |-  F/ x ps
2 ceqsalg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1605 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsalt 3118 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1315 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1381    = wceq 1383   F/wnf 1603    e. wcel 1804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-v 3097
This theorem is referenced by:  ceqsal  3122  uniiunlem  3573  ralrnmpt2  6402  sucprcregOLD  8029  fimaxre3  10499  pmapglbx  35233
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