MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbiegf Unicode version

Theorem csbiegf 3121
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 1533 . 2  |-  A. x
( x  =  A  ->  B  =  C )
3 csbiegf.1 . . 3  |-  ( A  e.  V  ->  F/_ x C )
4 csbiebt 3117 . . 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 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406   [_csb 3081
This theorem is referenced by:  csbief  3122  sbcco3g  3136  csbco3g  3138  fmptcof  5692  fmpt2co  6202  sumsn  12213  pcmpt  12940  elmptrab  17522  dvfsumrlim3  19380  itgsubstlem  19395  itgsubst  19396  ifeqeqx  23034  sbcung  23431  sbcopg  23433  sbcaltop  23926  bpolylem  24194  fprod1fi  24738  unirep  25761  monotuz  26438  oddcomabszz  26441  cdleme31so  29941  cdleme31sn  29942  cdleme31sn1  29943  cdleme31se  29944  cdleme31se2  29945  cdleme31sc  29946  cdleme31sde  29947  cdleme31sn2  29951  cdlemeg47rv2  30072  cdlemk41  30482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
  Copyright terms: Public domain W3C validator