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Theorem unab 3313
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 1905 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
2 df-clab 2104 . . 3  |-  ( y  e.  { x  |  ( ph  \/  ps ) }  <->  [ y  /  x ] ( ph  \/  ps ) )
3 df-clab 2104 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2104 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4orbi12i 738 . . 3  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  \/  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 212 . 2  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  \/  ps ) } )
76uneqri 3188 1  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 682    = wceq 1316    e. wcel 1465   [wsb 1720   {cab 2103    u. cun 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045
This theorem is referenced by:  unrab  3317  rabun2  3325  dfif6  3446  unopab  3977  dmun  4716  frecabex  6263
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