Definition df-if 4479

Description: Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4484 and iffalse 4487 for its values. In the 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".

An important use for us is in conjunction with the weak deduction theorem, which is described in the next section, beginning at dedth 4536. (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 4478	. 2 class if(𝜑, 𝐴, 𝐵)
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1540	. . . . . 6 class 𝑥
7	6, 2	wcel 2106	. . . . 5 wff 𝑥 ∈ 𝐴
8	7, 1	wa 397	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝜑)
9	6, 3	wcel 2106	. . . . 5 wff 𝑥 ∈ 𝐵
10	1	wn 3	. . . . 5 wff ¬ 𝜑
11	9, 10	wa 397	. . . 4 wff (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)
12	8, 11	wo 845	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
13	12, 5	cab 2714	. 2 class {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
14	4, 13	wceq 1541	1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}

This definition is referenced by:  dfif2  4480  dfif6  4481  iffalse  4487  rabsnifsb  4675  bj-df-ifc  34898