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Definition df-if 4526
Description: Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4531 and iffalse 4534 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 4584. (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 4525 . 2 class if(𝜑, 𝐴, 𝐵)
5 vx . . . . . . 7 setvar 𝑥
65cv 1539 . . . . . 6 class 𝑥
76, 2wcel 2108 . . . . 5 wff 𝑥𝐴
87, 1wa 395 . . . 4 wff (𝑥𝐴𝜑)
96, 3wcel 2108 . . . . 5 wff 𝑥𝐵
101wn 3 . . . . 5 wff ¬ 𝜑
119, 10wa 395 . . . 4 wff (𝑥𝐵 ∧ ¬ 𝜑)
128, 11wo 848 . . 3 wff ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))
1312, 5cab 2714 . 2 class {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
144, 13wceq 1540 1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
Colors of variables: wff setvar class
This definition is referenced by:  dfif2  4527  dfif6  4528  iffalse  4534  rabsnifsb  4722  bj-df-ifc  36581
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