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Theorem sbcco 3026
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 3013 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  ->  A  e.  _V )
2 sbcex 3013 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 dfsbcq 3006 . . 3  |-  ( z  =  A  ->  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. A  /  y ]. [. y  /  x ]. ph ) )
4 dfsbcq 3006 . . 3  |-  ( z  =  A  ->  ( [. z  /  x ]. ph  <->  [. A  /  x ]. ph ) )
5 sbsbc 3008 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
65sbbii 1643 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [. y  /  x ]. ph )
7 nfv 1609 . . . . . 6  |-  F/ y
ph
87sbco2 2039 . . . . 5  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
9 sbsbc 3008 . . . . 5  |-  ( [ z  /  y ]
[. y  /  x ]. ph  <->  [. z  /  y ]. [. y  /  x ]. ph )
106, 8, 93bitr3ri 267 . . . 4  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [ z  /  x ] ph )
11 sbsbc 3008 . . . 4  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
1210, 11bitri 240 . . 3  |-  ( [. z  /  y ]. [. y  /  x ]. ph  <->  [. z  /  x ]. ph )
133, 4, 12vtoclbg 2857 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
141, 2, 13pm5.21nii 342 1  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1638    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  sbc7  3031  sbccom  3075  sbcralt  3076  csbco  3103  aomclem6  27259  bnj62  29062  bnj610  29092  bnj976  29125  bnj1468  29194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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