Definition df-if 3700

Description: Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B." See iftrue 3705 and iffalse 3706 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 3740 for the main part of the weak deduction theorem, elimhyp 3747 to eliminate a hypothesis, and keephyp 3753 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

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	cif 3699	. 2 class if ( ph , A , B )
5		vx	. . . . . . 7 set x
6	5	cv 1648	. . . . . 6 class x
7	6, 2	wcel 1721	. . . . 5 wff x e. A
8	7, 1	wa 359	. . . 4 wff ( x e. A /\ ph )
9	6, 3	wcel 1721	. . . . 5 wff x e. B
10	1	wn 3	. . . . 5 wff -. ph
11	9, 10	wa 359	. . . 4 wff ( x e. B /\ -. ph )
12	8, 11	wo 358	. . 3 wff ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) )
13	12, 5	cab 2390	. 2 class { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
14	4, 13	wceq 1649	1 wff if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }

This definition is referenced by:  dfif2  3701  dfif6  3702  iffalse  3706