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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 3542 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | iffalse 3533 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶) | |
4 | iffalse 3533 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶) | |
5 | 3, 4 | eqtr4d 2206 | . . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
6 | 5 | adantl 275 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
7 | ifeq1dadc.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
8 | exmiddc 831 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
10 | 2, 6, 9 | mpjaodan 793 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 703 DECID wdc 829 = wceq 1348 ifcif 3525 |
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 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rab 2457 df-v 2732 df-un 3125 df-if 3526 |
This theorem is referenced by: sumeq2 11309 isumss 11341 prodeq2 11507 lgsval2lem 13626 lgsval4lem 13627 lgsneg 13640 lgsmod 13642 lgsdilem2 13652 |
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