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Theorem dfif2 3580
Description: An alternate definition of the conditional operator df-if 3579 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
Assertion
Ref Expression
dfif2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  B  ->  ph )  ->  ( x  e.  A  /\  ph ) ) }
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem dfif2
StepHypRef Expression
1 df-if 3579 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 df-or 359 . . . 4  |-  ( ( ( x  e.  B  /\  -.  ph )  \/  ( x  e.  A  /\  ph ) )  <->  ( -.  ( x  e.  B  /\  -.  ph )  -> 
( x  e.  A  /\  ph ) ) )
3 orcom 376 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) )  <->  ( (
x  e.  B  /\  -.  ph )  \/  (
x  e.  A  /\  ph ) ) )
4 iman 413 . . . . 5  |-  ( ( x  e.  B  ->  ph )  <->  -.  ( x  e.  B  /\  -.  ph ) )
54imbi1i 315 . . . 4  |-  ( ( ( x  e.  B  ->  ph )  ->  (
x  e.  A  /\  ph ) )  <->  ( -.  ( x  e.  B  /\  -.  ph )  -> 
( x  e.  A  /\  ph ) ) )
62, 3, 53bitr4i 268 . . 3  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) )  <->  ( (
x  e.  B  ->  ph )  ->  ( x  e.  A  /\  ph ) ) )
76abbii 2408 . 2  |-  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( x  e.  B  ->  ph )  ->  (
x  e.  A  /\  ph ) ) }
81, 7eqtri 2316 1  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  B  ->  ph )  ->  ( x  e.  A  /\  ph ) ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   ifcif 3578
This theorem is referenced by:  iftrue  3584  nfifd  3601
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-if 3579
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