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Theorem dfif6 3581
Description: An alternate definition of the conditional operator df-if 3579 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 3448 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  -.  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-rab 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2565 . . 3  |-  { x  e.  B  |  -.  ph }  =  { x  |  ( x  e.  B  /\  -.  ph ) }
42, 3uneq12i 3340 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  B  /\  -.  ph ) } )
5 df-if 3579 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
61, 4, 53eqtr4ri 2327 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 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    u. cun 3163   ifcif 3578
This theorem is referenced by:  ifeq1  3582  ifeq2  3583  dfif3  3588
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-rab 2565  df-v 2803  df-un 3170  df-if 3579
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