Definition df-if 4549

Description: Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4554 and iffalse 4557 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 4606. (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 4548	. 2 class if(𝜑, 𝐴, 𝐵)
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1536	. . . . . 6 class 𝑥
7	6, 2	wcel 2108	. . . . 5 wff 𝑥 ∈ 𝐴
8	7, 1	wa 395	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝜑)
9	6, 3	wcel 2108	. . . . 5 wff 𝑥 ∈ 𝐵
10	1	wn 3	. . . . 5 wff ¬ 𝜑
11	9, 10	wa 395	. . . 4 wff (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)
12	8, 11	wo 846	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
13	12, 5	cab 2717	. 2 class {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
14	4, 13	wceq 1537	1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}

This definition is referenced by:  dfif2  4550  dfif6  4551  iffalse  4557  rabsnifsb  4747  bj-df-ifc  36546