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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 3336 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | eqcomd 2045 | . . . . 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 148 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜃)) |
6 | 5 | 3ad2ant1 925 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 → 𝜃)) |
7 | iffalse 3339 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2045 | . . . . 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 148 | . . 3 ⊢ (𝜒 → (¬ 𝜑 → 𝜃)) |
12 | 11 | 3ad2ant2 926 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (¬ 𝜑 → 𝜃)) |
13 | exmiddc 744 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
14 | 13 | 3ad2ant3 927 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 ∨ ¬ 𝜑)) |
15 | 6, 12, 14 | mpjaod 638 | 1 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 629 DECID wdc 742 ∧ w3a 885 = wceq 1243 ifcif 3331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3an 887 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-if 3332 |
This theorem is referenced by: (None) |
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