Definition df-if 4376

Description: Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4381 and iffalse 4384 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 4431. (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 4375	. 2 class if(𝜑, 𝐴, 𝐵)
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1519	. . . . . 6 class 𝑥
7	6, 2	wcel 2079	. . . . 5 wff 𝑥 ∈ 𝐴
8	7, 1	wa 396	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝜑)
9	6, 3	wcel 2079	. . . . 5 wff 𝑥 ∈ 𝐵
10	1	wn 3	. . . . 5 wff ¬ 𝜑
11	9, 10	wa 396	. . . 4 wff (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)
12	8, 11	wo 842	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
13	12, 5	cab 2773	. 2 class {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
14	4, 13	wceq 1520	1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}

This definition is referenced by:  dfif2  4377  dfif6  4378  iffalse  4384  rabsnifsb  4559  bj-dfifc2  33463