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Theorem dfif6 3685
Description: An alternate definition of the conditional operator df-if 3683 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 3551 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  -.  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-rab 2658 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2658 . . 3  |-  { x  e.  B  |  -.  ph }  =  { x  |  ( x  e.  B  /\  -.  ph ) }
42, 3uneq12i 3442 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  B  /\  -.  ph ) } )
5 df-if 3683 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
61, 4, 53eqtr4ri 2418 1  |-  if (
ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   {crab 2653    u. cun 3261   ifcif 3682
This theorem is referenced by:  ifeq1  3686  ifeq2  3687  dfif3  3692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-un 3268  df-if 3683
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