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Theorem csbief 3445
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 3444 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
61, 5ax-mp 5 1  |-  [_ A  /  x ]_ B  =  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   F/_wnfc 2591   _Vcvv 3095   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-sbc 3314  df-csb 3421
This theorem is referenced by:  csbie  3446  csbun  3843  csbin  3846  csbingOLD  3847  csbif  3976  csbifgOLD  3977  csbopab  4769  csbopabgALT  4770  csbima12  5344  csbima12gOLD  5345  csbiota  5570  csbiotagOLD  5571  csbriota  6254  csbov123  6315  csbovgOLD  6317  pcmpt  14392  mpfrcl  18165  iundisj2  21936  iundisj2f  27425  iundisj2fi  27578  sbccom2f  30506  csbafv12g  32060  csbaovg  32103
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