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Theorem sbcco3g 3128
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 3124 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
2 elex 2762 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2332 . . . 4  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 3114 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
6 dfsbcq 2978 . . 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 188 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1363    e. wcel 2159   _Vcvv 2751   [.wsbc 2976   [_csb 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-sbc 2977  df-csb 3072
This theorem is referenced by:  fzshftral  10125
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