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Mirrors > Home > ILE Home > Th. List > ifcldadc | GIF version |
Description: Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
Ref | Expression |
---|---|
ifcldadc.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) |
ifcldadc.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) |
ifcldadc.dc | ⊢ (𝜑 → DECID 𝜓) |
Ref | Expression |
---|---|
ifcldadc | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3356 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 271 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifcldadc.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) | |
4 | 2, 3 | eqeltrd 2155 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
5 | iffalse 3359 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
6 | 5 | adantl 271 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
7 | ifcldadc.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) | |
8 | 6, 7 | eqeltrd 2155 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
9 | ifcldadc.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
10 | exmiddc 777 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
11 | 9, 10 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
12 | 4, 8, 11 | mpjaodan 744 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 661 DECID wdc 775 = wceq 1284 ∈ wcel 1433 ifcif 3351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-if 3352 |
This theorem is referenced by: eucalgval2 10435 lcmval 10445 |
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