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Mirrors > Home > MPE Home > Th. List > df-if | Structured version Visualization version GIF version |
Description: Definition of the
conditional operator for classes. The expression
if(𝜑,
𝐴, 𝐵) is read "if 𝜑 then
𝐴
else 𝐵". See
iftrue 4473 and iffalse 4476 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 4523. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
df-if | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | cA | . . 3 class 𝐴 | |
3 | cB | . . 3 class 𝐵 | |
4 | 1, 2, 3 | cif 4467 | . 2 class if(𝜑, 𝐴, 𝐵) |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1536 | . . . . . 6 class 𝑥 |
7 | 6, 2 | wcel 2114 | . . . . 5 wff 𝑥 ∈ 𝐴 |
8 | 7, 1 | wa 398 | . . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝜑) |
9 | 6, 3 | wcel 2114 | . . . . 5 wff 𝑥 ∈ 𝐵 |
10 | 1 | wn 3 | . . . . 5 wff ¬ 𝜑 |
11 | 9, 10 | wa 398 | . . . 4 wff (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) |
12 | 8, 11 | wo 843 | . . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) |
13 | 12, 5 | cab 2799 | . 2 class {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} |
14 | 4, 13 | wceq 1537 | 1 wff if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} |
Colors of variables: wff setvar class |
This definition is referenced by: dfif2 4469 dfif6 4470 iffalse 4476 rabsnifsb 4658 bj-df-ifc 33913 |
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