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Mirrors > Home > ILE Home > Th. List > ifbid | GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Ref | Expression |
---|---|
ifbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ifbid | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbid.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | ifbi 3369 | . 2 ⊢ ((𝜓 ↔ 𝜒) → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = 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-in1 576 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-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-if 3352 |
This theorem is referenced by: ifbieq1d 3371 ifbieq2d 3373 ifbieq12d 3375 sumeq1 10192 |
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