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Theorem unab 3374
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 1934 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
2 df-clab 2144 . . 3  |-  ( y  e.  { x  |  ( ph  \/  ps ) }  <->  [ y  /  x ] ( ph  \/  ps ) )
3 df-clab 2144 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2144 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4orbi12i 754 . . 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 3249 1  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 698    = wceq 1335   [wsb 1742    e. wcel 2128   {cab 2143    u. cun 3100
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106
This theorem is referenced by:  unrab  3378  rabun2  3386  dfif6  3507  unopab  4043  dmun  4793  frecabex  6345
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