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Theorem dfif6 3729
Description: An alternate definition of the conditional operator df-if 3727 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 3595 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  -.  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-rab 2701 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2701 . . 3  |-  { x  e.  B  |  -.  ph }  =  { x  |  ( x  e.  B  /\  -.  ph ) }
42, 3uneq12i 3486 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  B  /\  -.  ph ) } )
5 df-if 3727 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
61, 4, 53eqtr4ri 2461 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 1652    e. wcel 1725   {cab 2416   {crab 2696    u. cun 3305   ifcif 3726
This theorem is referenced by:  ifeq1  3730  ifeq2  3731  dfif3  3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-rab 2701  df-v 2945  df-un 3312  df-if 3727
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