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Theorem sbcco 2837
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 2824 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  ->  A  e.  _V )
2 sbcex 2824 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 dfsbcq 2818 . . 3  |-  ( z  =  A  ->  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. A  /  y ]. [. y  /  x ]. ph ) )
4 dfsbcq 2818 . . 3  |-  ( z  =  A  ->  ( [. z  /  x ]. ph  <->  [. A  /  x ]. ph ) )
5 sbsbc 2820 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
65sbbii 1689 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [. y  /  x ]. ph )
7 nfv 1462 . . . . . 6  |-  F/ y
ph
87sbco2 1881 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
9 sbsbc 2820 . . . . 5  |-  ( [ z  /  y ]
[. y  /  x ]. ph  <->  [. z  /  y ]. [. y  /  x ]. ph )
106, 8, 93bitr3ri 209 . . . 4  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [ z  /  x ] ph )
11 sbsbc 2820 . . . 4  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
1210, 11bitri 182 . . 3  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. z  /  x ]. ph )
133, 4, 12vtoclbg 2660 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
141, 2, 13pm5.21nii 653 1  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   [wsb 1686   _Vcvv 2602   [.wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  sbc7  2842  sbccom  2890  sbcralt  2891  sbcrext  2892  csbco  2918
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