![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ifbothdc | GIF version |
Description: A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.) |
Ref | Expression |
---|---|
ifbothdc.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) |
ifbothdc.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
ifbothdc | ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3484 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | eqcomd 2146 | . . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵)) |
3 | ifbothdc.1 | . . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
5 | 4 | biimpcd 158 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜃)) |
6 | 5 | 3ad2ant1 1003 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 → 𝜃)) |
7 | iffalse 3487 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2146 | . . . . 5 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵)) |
9 | ifbothdc.2 | . . . . 5 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜃)) |
11 | 10 | biimpcd 158 | . . 3 ⊢ (𝜒 → (¬ 𝜑 → 𝜃)) |
12 | 11 | 3ad2ant2 1004 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (¬ 𝜑 → 𝜃)) |
13 | exmiddc 822 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
14 | 13 | 3ad2ant3 1005 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 ∨ ¬ 𝜑)) |
15 | 6, 12, 14 | mpjaod 708 | 1 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 698 DECID wdc 820 ∧ w3a 963 = 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-3an 965 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-if 3480 |
This theorem is referenced by: isumlessdc 11297 |
Copyright terms: Public domain | W3C validator |