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Theorem sbcor 2954
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 2918 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps )  ->  A  e.  _V )
2 sbcex 2918 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 sbcex 2918 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
42, 3jaoi 706 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps )  ->  A  e.  _V )
5 dfsbcq2 2913 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
6 dfsbcq2 2913 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
7 dfsbcq2 2913 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
86, 7orbi12d 783 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
9 sbor 1928 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
105, 8, 9vtoclbg 2748 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
111, 4, 10pm5.21nii 694 1  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   [wsb 1736   _Vcvv 2687   [.wsbc 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-sbc 2911
This theorem is referenced by:  rabrsndc  3595
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