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Theorem sbcor 2867
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 2832 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps )  ->  A  e.  _V )
2 sbcex 2832 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 sbcex 2832 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
42, 3jaoi 669 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps )  ->  A  e.  _V )
5 dfsbcq2 2827 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
6 dfsbcq2 2827 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
7 dfsbcq2 2827 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
86, 7orbi12d 740 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
9 sbor 1871 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
105, 8, 9vtoclbg 2668 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
111, 4, 10pm5.21nii 653 1  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   [wsb 1687   _Vcvv 2610   [.wsbc 2824
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  rabrsndc  3478
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