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Theorem sbcco3g 2986
Description: Composition of two 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
sbcco3g  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
Distinct variable groups:    x, A    ph, x    x, C
Allowed substitution hints:    ph( y)    A( y)    B( x, y)    C( y)    V( x, y)

Proof of Theorem sbcco3g
StepHypRef Expression
1 sbcnestg 2982 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
2 elex 2631 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2230 . . . 4  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 2972 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
6 dfsbcq 2843 . . 3  |-  ( [_ A  /  x ]_ B  =  C  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  / 
y ]. ph ) )
72, 5, 63syl 17 . 2  |-  ( A  e.  V  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
81, 7bitrd 187 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2620   [.wsbc 2841   [_csb 2934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842  df-csb 2935
This theorem is referenced by:  fzshftral  9583
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