Definition df-if 3537

Description: Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵". See iftrue 3541 and iffalse 3544 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 the absence of excluded middle, this will tend to be useful where 𝜑 is decidable (in the sense of df-dc 835). (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 3536	. 2 class if(𝜑, 𝐴, 𝐵)
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1352	. . . . . 6 class 𝑥
7	6, 2	wcel 2148	. . . . 5 wff 𝑥 ∈ 𝐴
8	7, 1	wa 104	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝜑)
9	6, 3	wcel 2148	. . . . 5 wff 𝑥 ∈ 𝐵
10	1	wn 3	. . . . 5 wff ¬ 𝜑
11	9, 10	wa 104	. . . 4 wff (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)
12	8, 11	wo 708	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
13	12, 5	cab 2163	. 2 class {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
14	4, 13	wceq 1353	1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}

This definition is referenced by:  dfif6  3538  iftrue  3541  iffalse  3544  ifbi  3556  nfifd  3563  ifmdc  3576  if0ab  14596