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Mirrors > Home > ILE Home > Th. List > ifmdc | GIF version |
Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
ifmdc | ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2252 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ if(𝜑, 𝐵, 𝐶))) | |
2 | 1 | imbi1d 231 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) ↔ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)))) |
3 | df-if 3550 | . . . . . 6 ⊢ if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))} | |
4 | 3 | abeq2i 2300 | . . . . 5 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))) |
5 | simpr 110 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝜑) | |
6 | simpr 110 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ ¬ 𝜑) → ¬ 𝜑) | |
7 | 5, 6 | orim12i 760 | . . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑)) |
8 | 4, 7 | sylbi 121 | . . . 4 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) |
9 | 2, 8 | vtoclg 2812 | . . 3 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))) |
10 | 9 | pm2.43i 49 | . 2 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) |
11 | df-dc 836 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
12 | 10, 11 | sylibr 134 | 1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2160 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-if 3550 |
This theorem is referenced by: (None) |
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