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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 3356 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | eqcomd 2086 | . . . . 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 157 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜃)) |
6 | 5 | 3ad2ant1 959 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 → 𝜃)) |
7 | iffalse 3359 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2086 | . . . . 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 157 | . . 3 ⊢ (𝜒 → (¬ 𝜑 → 𝜃)) |
12 | 11 | 3ad2ant2 960 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (¬ 𝜑 → 𝜃)) |
13 | exmiddc 777 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
14 | 13 | 3ad2ant3 961 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 ∨ ¬ 𝜑)) |
15 | 6, 12, 14 | mpjaod 670 | 1 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∨ wo 661 DECID wdc 775 ∧ w3a 919 = wceq 1284 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-3an 921 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-if 3352 |
This theorem is referenced by: (None) |
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