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Definition df-if 3380
Description: Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3384 and iffalse 3387 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise."

In the absence of excluded middle, this will tend to be useful where  ph is decidable (in the sense of df-dc 779). (Contributed by NM, 15-May-1999.)

Assertion
Ref Expression
df-if  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
Distinct variable groups:    ph, x    x, A    x, B

Detailed syntax breakdown of Definition df-if
StepHypRef Expression
1 wph . . 3  wff  ph
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3cif 3379 . 2  class  if (
ph ,  A ,  B )
5 vx . . . . . . 7  setvar  x
65cv 1286 . . . . . 6  class  x
76, 2wcel 1436 . . . . 5  wff  x  e.  A
87, 1wa 102 . . . 4  wff  ( x  e.  A  /\  ph )
96, 3wcel 1436 . . . . 5  wff  x  e.  B
101wn 3 . . . . 5  wff  -.  ph
119, 10wa 102 . . . 4  wff  ( x  e.  B  /\  -.  ph )
128, 11wo 662 . . 3  wff  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )
1312, 5cab 2071 . 2  class  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
144, 13wceq 1287 1  wff  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
Colors of variables: wff set class
This definition is referenced by:  dfif6  3381  iftrue  3384  iffalse  3387  ifbi  3397  nfifd  3404  ifmdc  3414
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