Definition df-if 3664

Description: Define the conditional operator. Read if(φ, A, B) as "if φ then A else B." See iftrue 3669 and iffalse 3670 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 3704 for the main part of the weak deduction theorem, elimhyp 3711 to eliminate a hypothesis, and keephyp 3717 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(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}

Distinct variable groups:   φ,x   x,A   x,B

Detailed syntax breakdown of Definition df-if

Step	Hyp	Ref	Expression
1		wph	. . 3 wff φ
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	cif 3663	. 2 class if(φ, A, B)
5		vx	. . . . . . 7 setvar x
6	5	cv 1641	. . . . . 6 class x
7	6, 2	wcel 1710	. . . . 5 wff x ∈ A
8	7, 1	wa 358	. . . 4 wff (x ∈ A ∧ φ)
9	6, 3	wcel 1710	. . . . 5 wff x ∈ B
10	1	wn 3	. . . . 5 wff ¬ φ
11	9, 10	wa 358	. . . 4 wff (x ∈ B ∧ ¬ φ)
12	8, 11	wo 357	. . 3 wff ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))
13	12, 5	cab 2339	. 2 class {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
14	4, 13	wceq 1642	1 wff if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}

This definition is referenced by:  dfif2  3665  dfif6  3666  iffalse  3670