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

Theorem sbcorg 3206
Description: Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcorg  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )

Proof of Theorem sbcorg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3164 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  \/  ps )  <->  [. A  /  x ]. ( ph  \/  ps ) ) )
2 dfsbcq2 3164 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3164 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3orbi12d 691 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  \/  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
5 sbor 2138 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
61, 4, 5vtoclbg 3012 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1652   [wsb 1658    e. wcel 1725   [.wsbc 3161
This theorem is referenced by:  sbc3org  28616  sbc3orgVD  28963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
  Copyright terms: Public domain W3C validator