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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 3388 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | iffalse 3381 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶) | |
4 | iffalse 3381 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶) | |
5 | 3, 4 | eqtr4d 2118 | . . 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 778 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
10 | 2, 6, 9 | mpjaodan 745 | 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 662 DECID wdc 776 = wceq 1285 ifcif 3373 |
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 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rab 2362 df-v 2614 df-un 2988 df-if 3374 |
This theorem is referenced by: sumeq2d 10568 sumeq2 10569 |
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