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

Theorem csbiegf 3283
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 1555 . 2  |-  A. x
( x  =  A  ->  B  =  C )
3 csbiegf.1 . . 3  |-  ( A  e.  V  ->  F/_ x C )
4 csbiebt 3279 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
53, 4mpdan 650 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
62, 5mpbii 203 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   F/_wnfc 2558   [_csb 3243
This theorem is referenced by:  csbief  3284  sbcco3g  3297  csbco3g  3299  fmptcof  5893  fmpt2co  6421  sumsn  12522  pcmpt  13249  elmptrab  17847  dvfsumrlim3  19905  itgsubstlem  19920  itgsubst  19921  ifeqeqx  23989  sbcung  25112  sbcopg  25114  prodsn  25275  sbcaltop  25774  bpolylem  26042  unirep  26351  monotuz  26941  oddcomabszz  26944  cdleme31so  31015  cdleme31sn  31016  cdleme31sn1  31017  cdleme31se  31018  cdleme31se2  31019  cdleme31sc  31020  cdleme31sde  31021  cdleme31sn2  31025  cdlemeg47rv2  31146  cdlemk41  31556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
  Copyright terms: Public domain W3C validator