Definition df-if 3563

Description: Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B". See iftrue 3567 and iffalse 3570 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 836). (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

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	cif 3562	. 2 class if ( ph , A , B )
5		vx	. . . . . . 7 setvar x
6	5	cv 1363	. . . . . 6 class x
7	6, 2	wcel 2167	. . . . 5 wff x e. A
8	7, 1	wa 104	. . . 4 wff ( x e. A /\ ph )
9	6, 3	wcel 2167	. . . . 5 wff x e. B
10	1	wn 3	. . . . 5 wff -. ph
11	9, 10	wa 104	. . . 4 wff ( x e. B /\ -. ph )
12	8, 11	wo 709	. . 3 wff ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) )
13	12, 5	cab 2182	. 2 class { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
14	4, 13	wceq 1364	1 wff if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }

This definition is referenced by:  dfif6  3564  iftrue  3567  iffalse  3570  ifbi  3582  nfifd  3589  ifmdc  3602  if0ab  15535