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Theorem sbcco 3185
Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcco  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Distinct variable group:    ph, y
Allowed substitution hints:    ph( x)    A( x, y)

Proof of Theorem sbcco
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3172 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  ->  A  e.  _V )
2 sbcex 3172 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 dfsbcq 3165 . . 3  |-  ( z  =  A  ->  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. A  /  y ]. [. y  /  x ]. ph ) )
4 dfsbcq 3165 . . 3  |-  ( z  =  A  ->  ( [. z  /  x ]. ph  <->  [. A  /  x ]. ph ) )
5 sbsbc 3167 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
65sbbii 1666 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [. y  /  x ]. ph )
7 nfv 1630 . . . . . 6  |-  F/ y
ph
87sbco2 2163 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
9 sbsbc 3167 . . . . 5  |-  ( [ z  /  y ]
[. y  /  x ]. ph  <->  [. z  /  y ]. [. y  /  x ]. ph )
106, 8, 93bitr3ri 269 . . . 4  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [ z  /  x ] ph )
11 sbsbc 3167 . . . 4  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
1210, 11bitri 242 . . 3  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. z  /  x ]. ph )
133, 4, 12vtoclbg 3014 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
141, 2, 13pm5.21nii 344 1  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1659    e. wcel 1726   _Vcvv 2958   [.wsbc 3163
This theorem is referenced by:  sbc7  3190  sbccom  3234  sbcralt  3235  csbco  3262  aomclem6  27148  bnj62  29159  bnj610  29189  bnj976  29222  bnj1468  29291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-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
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