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Theorem csbiegf 3459
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiegf.1  |-  ( A  e.  V  ->  F/_ x C )
csbiegf.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
csbiegf  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem csbiegf
StepHypRef Expression
1 csbiegf.2 . . 3  |-  ( x  =  A  ->  B  =  C )
21ax-gen 1601 . 2  |-  A. x
( x  =  A  ->  B  =  C )
3 csbiegf.1 . . 3  |-  ( A  e.  V  ->  F/_ x C )
4 csbiebt 3455 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
53, 4mpdan 668 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
62, 5mpbii 211 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   F/_wnfc 2615   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436
This theorem is referenced by:  csbief  3460  sbcco3g  3843  csbco3g  3844  csbungOLD  3858  csbopg  4231  fmptcof  6053  fmpt2co  6863  sumsn  13522  pcmpt  14266  chfacfpmmulfsupp  19131  elmptrab  20063  dvfsumrlim3  22169  itgsubstlem  22184  itgsubst  22185  nbgraopALT  24100  ifeqeqx  27093  disjunsn  27126  prodsn  28669  sbcaltop  29208  bpolylem  29387  unirep  29806  monotuz  30481  oddcomabszz  30484  cdleme31so  35175  cdleme31sn  35176  cdleme31sn1  35177  cdleme31se  35178  cdleme31se2  35179  cdleme31sc  35180  cdleme31sde  35181  cdleme31sn2  35185  cdlemeg47rv2  35306  cdlemk41  35716
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