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Mirrors > Home > ILE Home > Th. List > ifbieq1d | GIF version |
Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
Ref | Expression |
---|---|
ifbieq1d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
ifbieq1d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifbieq1d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq1d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | ifbid 3412 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐴, 𝐶)) |
3 | ifbieq1d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq1d 3408 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
5 | 2, 4 | eqtrd 2120 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
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-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 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-clel 2084 df-nfc 2217 df-rab 2368 df-v 2621 df-un 3003 df-if 3394 |
This theorem is referenced by: iseqf1olemfvp 9926 seq3f1olemqsum 9929 seq3f1oleml 9932 seq3f1o 9933 bcval 10157 isumrblem 10765 isummolem3 10770 isummolem2a 10771 isummo 10773 zisum 10774 fisum 10778 fsum3 10779 isumss 10783 isumss2 10785 fisumcvg2 10786 fsum3cvg2 10787 fisumser 10790 fsumcl2lem 10792 fsumadd 10800 sumsnf 10803 fsummulc2 10842 isumlessdc 10890 |
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