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Theorem csbco3g 3151
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 3144 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ D  =  [_ [_ A  /  x ]_ B  /  y ]_ D )
2 elex 2809 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2433 . . . . 5  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 3134 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
62, 5syl 15 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  C )
76csbeq1d 3100 . 2  |-  ( A  e.  V  ->  [_ [_ A  /  x ]_ B  / 
y ]_ D  =  [_ C  /  y ]_ D
)
81, 7eqtrd 2328 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ D  =  [_ C  /  y ]_ D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094
This theorem is referenced by:  csbco3gOLD  3152
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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