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Mirrors > Home > ILE Home > Th. List > ifbothdadc | GIF version |
Description: A formula 𝜃 containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.) |
Ref | Expression |
---|---|
ifbothdc.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) |
ifbothdc.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) |
ifbothdadc.3 | ⊢ ((𝜂 ∧ 𝜑) → 𝜓) |
ifbothdadc.4 | ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) |
ifbothdadc.dc | ⊢ (𝜂 → DECID 𝜑) |
Ref | Expression |
---|---|
ifbothdadc | ⊢ (𝜂 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbothdadc.3 | . . 3 ⊢ ((𝜂 ∧ 𝜑) → 𝜓) | |
2 | iftrue 3484 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | 2 | eqcomd 2146 | . . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵)) |
4 | ifbothdc.1 | . . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
6 | 5 | adantl 275 | . . 3 ⊢ ((𝜂 ∧ 𝜑) → (𝜓 ↔ 𝜃)) |
7 | 1, 6 | mpbid 146 | . 2 ⊢ ((𝜂 ∧ 𝜑) → 𝜃) |
8 | ifbothdadc.4 | . . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) | |
9 | iffalse 3487 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
10 | 9 | eqcomd 2146 | . . . . 5 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵)) |
11 | ifbothdc.2 | . . . . 5 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) | |
12 | 10, 11 | syl 14 | . . . 4 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜃)) |
13 | 12 | adantl 275 | . . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → (𝜒 ↔ 𝜃)) |
14 | 8, 13 | mpbid 146 | . 2 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜃) |
15 | ifbothdadc.dc | . . 3 ⊢ (𝜂 → DECID 𝜑) | |
16 | exmiddc 822 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
17 | 15, 16 | syl 14 | . 2 ⊢ (𝜂 → (𝜑 ∨ ¬ 𝜑)) |
18 | 7, 14, 17 | mpjaodan 788 | 1 ⊢ (𝜂 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 820 = wceq 1332 ifcif 3479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-if 3480 |
This theorem is referenced by: hashgcdeq 11940 |
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