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Definition df-if 3374
Description: Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵." See iftrue 3378 and iffalse 3381 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 777). (Contributed by NM, 15-May-1999.)

Assertion
Ref Expression
df-if if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-if
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3cif 3373 . 2 class if(𝜑, 𝐴, 𝐵)
5 vx . . . . . . 7 setvar 𝑥
65cv 1284 . . . . . 6 class 𝑥
76, 2wcel 1434 . . . . 5 wff 𝑥𝐴
87, 1wa 102 . . . 4 wff (𝑥𝐴𝜑)
96, 3wcel 1434 . . . . 5 wff 𝑥𝐵
101wn 3 . . . . 5 wff ¬ 𝜑
119, 10wa 102 . . . 4 wff (𝑥𝐵 ∧ ¬ 𝜑)
128, 11wo 662 . . 3 wff ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))
1312, 5cab 2069 . 2 class {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
144, 13wceq 1285 1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
Colors of variables: wff set class
This definition is referenced by:  dfif6  3375  iftrue  3378  iffalse  3381  ifbi  3391  nfifd  3398
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