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Theorem unab 3448
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 2019 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
2 df-clab 2283 . . 3  |-  ( y  e.  { x  |  ( ph  \/  ps ) }  <->  [ y  /  x ] ( ph  \/  ps ) )
3 df-clab 2283 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2283 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4orbi12i 507 . . 3  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  \/  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 269 . 2  |-  ( ( y  e.  { x  |  ph }  \/  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  \/  ps ) } )
76uneqri 3330 1  |-  ( { x  |  ph }  u.  { x  |  ps } )  =  {
x  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282    u. cun 3163
This theorem is referenced by:  unrab  3452  rabun2  3460  dfif6  3581  unopab  4111  dmun  4901  hashf1lem2  11410  vdwlem6  13049  vdgrun  23908  clsbldimp  25191  diophun  26956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
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