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Mirrors > Home > ILE Home > Th. List > ifeq1dadc | GIF version |
Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
ifeq1dadc.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) |
ifeq1dadc.dc | ⊢ (𝜑 → DECID 𝜓) |
Ref | Expression |
---|---|
ifeq1dadc | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1dadc.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) | |
2 | 1 | ifeq1d 3366 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | iffalse 3359 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶) | |
4 | iffalse 3359 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶) | |
5 | 3, 4 | eqtr4d 2116 | . . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
6 | 5 | adantl 271 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
7 | ifeq1dadc.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
8 | exmiddc 777 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
10 | 2, 6, 9 | mpjaodan 744 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 661 DECID wdc 775 = 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-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 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-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-v 2603 df-un 2977 df-if 3352 |
This theorem is referenced by: (None) |
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