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Theorem sbcco3g 3136
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 3130 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
2 elex 2796 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2420 . . . 4  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 3121 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
6 dfsbcq 2993 . . 3  |-  ( [_ A  /  x ]_ B  =  C  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  / 
y ]. ph ) )
72, 5, 63syl 18 . 2  |-  ( A  e.  V  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
81, 7bitrd 244 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  sbcco3gOLD  3137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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