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Theorem csbiegf 3134
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 1536 . 2  |-  A. x
( x  =  A  ->  B  =  C )
3 csbiegf.1 . . 3  |-  ( A  e.  V  ->  F/_ x C )
4 csbiebt 3130 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
53, 4mpdan 649 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
62, 5mpbii 202 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   F/_wnfc 2419   [_csb 3094
This theorem is referenced by:  csbief  3135  sbcco3g  3149  csbco3g  3151  fmptcof  5708  fmpt2co  6218  sumsn  12229  pcmpt  12956  elmptrab  17538  dvfsumrlim3  19396  itgsubstlem  19411  itgsubst  19412  ifeqeqx  23050  sbcung  24035  sbcopg  24037  sbcaltop  24587  bpolylem  24855  fprod1fi  25429  unirep  26452  monotuz  27129  oddcomabszz  27132  cdleme31so  31190  cdleme31sn  31191  cdleme31sn1  31192  cdleme31se  31193  cdleme31se2  31194  cdleme31sc  31195  cdleme31sde  31196  cdleme31sn2  31200  cdlemeg47rv2  31321  cdlemk41  31731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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