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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 3570 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐴, 𝐶)) |
3 | ifbieq1d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq1d 3566 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
5 | 2, 4 | eqtrd 2222 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-if 3550 |
This theorem is referenced by: ctssdclemn0 7140 ctssdc 7143 enumctlemm 7144 iseqf1olemfvp 10530 seq3f1olemqsum 10533 seq3f1oleml 10536 seq3f1o 10537 bcval 10764 sumrbdclem 11420 summodclem3 11423 summodclem2a 11424 summodc 11426 zsumdc 11427 fsum3 11430 isumss 11434 isumss2 11436 fsum3cvg2 11437 fsum3ser 11440 fsumcl2lem 11441 fsumadd 11449 sumsnf 11452 fsummulc2 11491 isumlessdc 11539 cbvprod 11601 prodrbdclem 11614 prodmodclem3 11618 prodmodclem2a 11619 prodmodc 11621 zproddc 11622 fprodseq 11626 fprodntrivap 11627 prodssdc 11632 fprodmul 11634 prodsnf 11635 pcmpt 12378 pcmptdvds 12380 lgsval 14883 lgsfvalg 14884 lgsdir 14914 lgsdilem2 14915 lgsdi 14916 lgsne0 14917 |
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