MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcco3g Structured version   Unicode version

Theorem sbcco3g 3307
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 3302 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
2 elex 2966 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 nfcvd 2575 . . . 4  |-  ( A  e.  _V  ->  F/_ x C )
4 sbcco3g.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
53, 4csbiegf 3293 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  C )
6 dfsbcq 3165 . . 3  |-  ( [_ A  /  x ]_ B  =  C  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  / 
y ]. ph ) )
72, 5, 63syl 19 . 2  |-  ( A  e.  V  ->  ( [. [_ A  /  x ]_ B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
81, 7bitrd 246 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  sbcco3gOLD  3308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
  Copyright terms: Public domain W3C validator