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Theorem sbcco 3350
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 3337 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  ->  A  e.  _V )
2 sbcex 3337 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 dfsbcq 3329 . . 3  |-  ( z  =  A  ->  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. A  /  y ]. [. y  /  x ]. ph ) )
4 dfsbcq 3329 . . 3  |-  ( z  =  A  ->  ( [. z  /  x ]. ph  <->  [. A  /  x ]. ph ) )
5 sbsbc 3331 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
65sbbii 1747 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [. y  /  x ]. ph )
7 nfv 1708 . . . . . 6  |-  F/ y
ph
87sbco2 2159 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
9 sbsbc 3331 . . . . 5  |-  ( [ z  /  y ]
[. y  /  x ]. ph  <->  [. z  /  y ]. [. y  /  x ]. ph )
106, 8, 93bitr3ri 276 . . . 4  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [ z  /  x ] ph )
11 sbsbc 3331 . . . 4  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
1210, 11bitri 249 . . 3  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. z  /  x ]. ph )
133, 4, 12vtoclbg 3168 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
141, 2, 13pm5.21nii 353 1  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1740    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  sbc7  3355  sbccom  3405  sbcralt  3406  csbco  3440  sbccom2  30735  sbccom2f  30736  aomclem6  31209  bnj62  33916  bnj610  33947  bnj976  33979  bnj1468  34047
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