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Theorem unab 3266
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 1876 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
2 df-clab 2075 . . 3  |-  ( y  e.  { x  |  ( ph  \/  ps ) }  <->  [ y  /  x ] ( ph  \/  ps ) )
3 df-clab 2075 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2075 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4orbi12i 716 . . 3  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  \/  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 211 . 2  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  \/  ps ) } )
76uneqri 3142 1  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 664    = wceq 1289    e. wcel 1438   [wsb 1692   {cab 2074    u. cun 2997
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003
This theorem is referenced by:  unrab  3270  rabun2  3278  dfif6  3395  unopab  3917  dmun  4643  frecabex  6163
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