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Definition df-if 3526
Description: Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵". See iftrue 3530 and iffalse 3533 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 830). (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 3525 . 2 class if(𝜑, 𝐴, 𝐵)
5 vx . . . . . . 7 setvar 𝑥
65cv 1347 . . . . . 6 class 𝑥
76, 2wcel 2141 . . . . 5 wff 𝑥𝐴
87, 1wa 103 . . . 4 wff (𝑥𝐴𝜑)
96, 3wcel 2141 . . . . 5 wff 𝑥𝐵
101wn 3 . . . . 5 wff ¬ 𝜑
119, 10wa 103 . . . 4 wff (𝑥𝐵 ∧ ¬ 𝜑)
128, 11wo 703 . . 3 wff ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))
1312, 5cab 2156 . 2 class {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
144, 13wceq 1348 1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
Colors of variables: wff set class
This definition is referenced by:  dfif6  3527  iftrue  3530  iffalse  3533  ifbi  3545  nfifd  3552  ifmdc  3563  if0ab  13762
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