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Theorem csbief 3235
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbief.1  |-  A  e. 
_V
csbief.2  |-  F/_ x C
csbief.3  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
csbief  |-  [_ A  /  x ]_ B  =  C
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem csbief
StepHypRef Expression
1 csbief.1 . 2  |-  A  e. 
_V
2 csbief.2 . . . 4  |-  F/_ x C
32a1i 11 . . 3  |-  ( A  e.  _V  ->  F/_ x C )
4 csbief.3 . . 3  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 3234 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
61, 5ax-mp 8 1  |-  [_ A  /  x ]_ B  =  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   F/_wnfc 2510   _Vcvv 2899   [_csb 3194
This theorem is referenced by:  csbing  3491  csbifg  3710  csbopabg  4224  pofun  4460  csbima12g  5153  csbiotag  5387  csbovg  6051  csbriotag  6498  eqerlem  6873  fsum  12441  fsumcnv  12484  fsumshftm  12491  fsum0diag2  12493  ruclem1  12757  pcmpt  13188  odval  15099  psrass1lem  16369  iundisj2  19310  isibl  19524  dfitg  19528  dvfsumlem2  19778  mpfrcl  19806  fsumdvdsmul  20847  iundisj2f  23873  iundisj2fi  23991  fprod  25046  bpolyval  25809  fphpd  26568  monotuz  26695  oddcomabszz  26698  fnwe2val  26815  fnwe2lem1  26816  mamufval  27112  csbafv12g  27670  csbaovg  27713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105  df-csb 3195
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