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Theorem sbcor 3372
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
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 3337 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps )  ->  A  e.  _V )
2 sbcex 3337 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 sbcex 3337 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
42, 3jaoi 379 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps )  ->  A  e.  _V )
5 dfsbcq2 3330 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
6 dfsbcq2 3330 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
7 dfsbcq2 3330 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
86, 7orbi12d 709 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
9 sbor 2140 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
105, 8, 9vtoclbg 3168 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
111, 4, 10pm5.21nii 353 1  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1395   [wsb 1740    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  sbcori  30716  sbc3or  33445
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