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Theorem dfif6 3887
Description: An alternate definition of the conditional operator df-if 3885 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 3716 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  -.  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-rab 2762 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2762 . . 3  |-  { x  e.  B  |  -.  ph }  =  { x  |  ( x  e.  B  /\  -.  ph ) }
42, 3uneq12i 3594 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  B  /\  -.  ph ) } )
5 df-if 3885 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
61, 4, 53eqtr4ri 2442 1  |-  if (
ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2757    u. cun 3411   ifcif 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-un 3418  df-if 3885
This theorem is referenced by:  ifeq1  3888  ifeq2  3889  dfif3  3898
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