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Definition df-if 3326
Description: Define the conditional operator. Read if(φ, A, B) as "if φ then A else B." See iftrue 3330 and iffalse 3333 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 742). (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
StepHypRef Expression
1 wph . . 3 wff φ
2 cA . . 3 class A
3 cB . . 3 class B
41, 2, 3cif 3325 . 2 class if(φ, A, B)
5 vx . . . . . . 7 setvar x
65cv 1241 . . . . . 6 class x
76, 2wcel 1390 . . . . 5 wff x A
87, 1wa 97 . . . 4 wff (x A φ)
96, 3wcel 1390 . . . . 5 wff x B
101wn 3 . . . . 5 wff ¬ φ
119, 10wa 97 . . . 4 wff (x B ¬ φ)
128, 11wo 628 . . 3 wff ((x A φ) (x B ¬ φ))
1312, 5cab 2023 . 2 class {x ∣ ((x A φ) (x B ¬ φ))}
144, 13wceq 1242 1 wff if(φ, A, B) = {x ∣ ((x A φ) (x B ¬ φ))}
Colors of variables: wff set class
This definition is referenced by:  dfif6  3327  iftrue  3330  iffalse  3333  ifbi  3342  nfifd  3349
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