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Theorem csbiegf 3293
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 1556 . 2  |-  A. x
( x  =  A  ->  B  =  C )
3 csbiegf.1 . . 3  |-  ( A  e.  V  ->  F/_ x C )
4 csbiebt 3289 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
53, 4mpdan 651 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
62, 5mpbii 204 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   F/_wnfc 2561   [_csb 3253
This theorem is referenced by:  csbief  3294  sbcco3g  3307  csbco3g  3309  fmptcof  5905  fmpt2co  6433  sumsn  12539  pcmpt  13266  elmptrab  17864  dvfsumrlim3  19922  itgsubstlem  19937  itgsubst  19938  ifeqeqx  24006  sbcung  25129  sbcopg  25131  prodsn  25291  sbcaltop  25831  bpolylem  26099  unirep  26428  monotuz  27018  oddcomabszz  27021  cdleme31so  31250  cdleme31sn  31251  cdleme31sn1  31252  cdleme31se  31253  cdleme31se2  31254  cdleme31sc  31255  cdleme31sde  31256  cdleme31sn2  31260  cdlemeg47rv2  31381  cdlemk41  31791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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