MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcor Structured version   Visualization version   Unicode version

Theorem sbcor 3323
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 3289 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps )  ->  A  e.  _V )
2 sbcex 3289 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
3 sbcex 3289 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
42, 3jaoi 385 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps )  ->  A  e.  _V )
5 dfsbcq2 3282 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
6 dfsbcq2 3282 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
7 dfsbcq2 3282 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
86, 7orbi12d 721 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
9 sbor 2238 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
105, 8, 9vtoclbg 3120 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
111, 4, 10pm5.21nii 359 1  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    \/ wo 374    = wceq 1455   [wsb 1808    e. wcel 1898   _Vcvv 3057   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280
This theorem is referenced by:  sbcori  32391  sbc3or  36933
  Copyright terms: Public domain W3C validator