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Definition df-if 3579
Description: Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3584 and iffalse 3585 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 older versions of this database, this operator was denoted "ded" and called the "deduction class.")

An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role,  A is a class variable in the hypothesis and  B is a class (usually a constant) that makes the hypothesis true when it is substituted for  A. See dedth 3619 for the main part of the weak deduction theorem, elimhyp 3626 to eliminate a hypothesis, and keephyp 3632 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (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 3578 . 2  class  if (
ph ,  A ,  B )
5 vx . . . . . . 7  set  x
65cv 1631 . . . . . 6  class  x
76, 2wcel 1696 . . . . 5  wff  x  e.  A
87, 1wa 358 . . . 4  wff  ( x  e.  A  /\  ph )
96, 3wcel 1696 . . . . 5  wff  x  e.  B
101wn 3 . . . . 5  wff  -.  ph
119, 10wa 358 . . . 4  wff  ( x  e.  B  /\  -.  ph )
128, 11wo 357 . . 3  wff  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )
1312, 5cab 2282 . 2  class  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
144, 13wceq 1632 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:  dfif2  3580  dfif6  3581  iffalse  3585
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