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Theorem dfif6 3571
Description: An alternate definition of the conditional operator df-if 3569 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfif6  |-  if (
ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem dfif6
StepHypRef Expression
1 unab 3438 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  -.  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-rab 2555 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2555 . . 3  |-  { x  e.  B  |  -.  ph }  =  { x  |  ( x  e.  B  /\  -.  ph ) }
42, 3uneq12i 3330 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  B  /\  -.  ph ) } )
5 df-if 3569 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
61, 4, 53eqtr4ri 2317 1  |-  if (
ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 5    \/ wo 359    /\ wa 360    = wceq 1625    e. wcel 1687   {cab 2272   {crab 2550    u. cun 3153   ifcif 3568
This theorem is referenced by:  ifeq1  3572  ifeq2  3573  dfif3  3578
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-rab 2555  df-v 2793  df-un 3160  df-if 3569
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