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Theorem sbcco 3175
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 3162 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  ->  A  e.  _V )
2 sbcex 3162 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 dfsbcq 3155 . . 3  |-  ( z  =  A  ->  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. A  /  y ]. [. y  /  x ]. ph ) )
4 dfsbcq 3155 . . 3  |-  ( z  =  A  ->  ( [. z  /  x ]. ph  <->  [. A  /  x ]. ph ) )
5 sbsbc 3157 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
65sbbii 1665 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [. y  /  x ]. ph )
7 nfv 1629 . . . . . 6  |-  F/ y
ph
87sbco2 2161 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
9 sbsbc 3157 . . . . 5  |-  ( [ z  /  y ]
[. y  /  x ]. ph  <->  [. z  /  y ]. [. y  /  x ]. ph )
106, 8, 93bitr3ri 268 . . . 4  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [ z  /  x ] ph )
11 sbsbc 3157 . . . 4  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
1210, 11bitri 241 . . 3  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. z  /  x ]. ph )
133, 4, 12vtoclbg 3004 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
141, 2, 13pm5.21nii 343 1  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658    e. wcel 1725   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  sbc7  3180  sbccom  3224  sbcralt  3225  csbco  3252  aomclem6  27125  bnj62  29022  bnj610  29052  bnj976  29085  bnj1468  29154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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