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Definition df-if 4522
Description: Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4527 and iffalse 4530 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 4579. (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 4521 . 2 class if(𝜑, 𝐴, 𝐵)
5 vx . . . . . . 7 setvar 𝑥
65cv 1532 . . . . . 6 class 𝑥
76, 2wcel 2098 . . . . 5 wff 𝑥𝐴
87, 1wa 395 . . . 4 wff (𝑥𝐴𝜑)
96, 3wcel 2098 . . . . 5 wff 𝑥𝐵
101wn 3 . . . . 5 wff ¬ 𝜑
119, 10wa 395 . . . 4 wff (𝑥𝐵 ∧ ¬ 𝜑)
128, 11wo 844 . . 3 wff ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))
1312, 5cab 2701 . 2 class {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
144, 13wceq 1533 1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
Colors of variables: wff setvar class
This definition is referenced by:  dfif2  4523  dfif6  4524  iffalse  4530  rabsnifsb  4719  bj-df-ifc  35948
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