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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 3498 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐴, 𝐶)) |
3 | ifbieq1d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq1d 3494 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
5 | 2, 4 | eqtrd 2173 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ifcif 3479 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-un 3080 df-if 3480 |
This theorem is referenced by: ctssdclemn0 7003 ctssdc 7006 enumctlemm 7007 iseqf1olemfvp 10301 seq3f1olemqsum 10304 seq3f1oleml 10307 seq3f1o 10308 bcval 10527 sumrbdclem 11178 summodclem3 11181 summodclem2a 11182 summodc 11184 zsumdc 11185 fsum3 11188 isumss 11192 isumss2 11194 fsum3cvg2 11195 fsum3ser 11198 fsumcl2lem 11199 fsumadd 11207 sumsnf 11210 fsummulc2 11249 isumlessdc 11297 cbvprod 11359 prodrbdclem 11372 prodmodclem3 11376 prodmodclem2a 11377 prodmodc 11379 zproddc 11380 fprodseq 11384 |
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