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Theorem sbcor 3069
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcor  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )

Proof of Theorem sbcor
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcex 3034 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps )  ->  A  e.  _V )
2 sbcex 3034 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 sbcex 3034 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
42, 3jaoi 368 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps )  ->  A  e.  _V )
5 dfsbcq2 3028 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
6 dfsbcq2 3028 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
7 dfsbcq2 3028 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
86, 7orbi12d 690 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
9 sbor 2038 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
105, 8, 9vtoclbg 2878 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
111, 4, 10pm5.21nii 342 1  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1633   [wsb 1639    e. wcel 1701   _Vcvv 2822   [.wsbc 3025
This theorem is referenced by:  rabrsn  27231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026
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