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Theorem unab 3544
Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unab  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }

Proof of Theorem unab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbor 2092 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
2 df-clab 2367 . . 3  |-  ( y  e.  { x  |  ( ph  \/  ps ) }  <->  [ y  /  x ] ( ph  \/  ps ) )
3 df-clab 2367 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2367 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4orbi12i 508 . . 3  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  \/  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 270 . 2  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  \/  ps ) } )
76uneqri 3425 1  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1649   [wsb 1655    e. wcel 1717   {cab 2366    u. cun 3254
This theorem is referenced by:  unrab  3548  rabun2  3556  dfif6  3678  unopab  4218  dmun  5009  hashf1lem2  11625  vdwlem6  13274  vdgrun  21513  vdgrfiun  21514  diophun  26516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-un 3261
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