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Mirrors > Home > ILE Home > Th. List > ifeq1d | GIF version |
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Ref | Expression |
---|---|
ifeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq1d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ifeq1 3477 | . 2 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ifcif 3474 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-un 3075 df-if 3475 |
This theorem is referenced by: ifeq12d 3491 ifbieq1d 3494 ifeq1dadc 3502 iseqf1olemjpcl 10268 iseqf1olemqpcl 10269 iseqf1olemfvp 10270 seq3f1olemqsum 10273 seq3f1olemp 10275 summodc 11152 fsum3 11156 fsum3ser 11166 isumlessdc 11265 prodeq2w 11325 prodmodc 11347 |
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