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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 3411 | . 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 1289 ifcif 3393 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-11 1442 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-if 3394 |
This theorem is referenced by: ifbieq1d 3413 ifbieq2d 3415 ifbieq12d 3417 ifandc 3427 fodjuomnilemm 6799 fodjuomnilem0 6800 fodjuomni 6802 nnnninf 6804 0tonninf 9841 1tonninf 9842 sumeq1 10740 isummo 10769 zisum 10770 fisum 10774 isumss 10779 sumsplitdc 10822 nninfalllemn 11853 nninfalllem1 11854 nninfsellemdc 11857 nninfself 11860 nninfsellemeq 11861 nninfsellemqall 11862 nninfsellemeqinf 11863 nninfomni 11866 |
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