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Theorem csbco3g 2986
Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
csbco3g  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ D  =  [_ C  /  y ]_ D )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    A( y)    B( x, y)    C( y)    D( y)    V( x, y)

Proof of Theorem csbco3g
StepHypRef Expression
1 csbnestg 2982 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ D  =  [_ [_ A  /  x ]_ B  /  y ]_ D )
2 elex 2630 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2229 . . . . 5  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 2971 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
62, 5syl 14 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
76csbeq1d 2939 . 2  |-  ( A  e.  V  ->  [_ [_ A  /  x ]_ B  / 
y ]_ D  =  [_ C  /  y ]_ D
)
81, 7eqtrd 2120 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ D  =  [_ C  /  y ]_ D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by: (None)
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