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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 3402 | . 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 1290 ifcif 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rab 2369 df-v 2624 df-un 3006 df-if 3400 |
This theorem is referenced by: ifeq12d 3416 ifbieq1d 3419 ifeq1dadc 3427 iseqf1olemjpcl 9987 iseqf1olemqpcl 9988 iseqf1olemfvp 9989 seq3f1olemqsum 9992 seq3f1olemp 9994 isummo 10836 fisum 10841 fisumser 10853 isumlessdc 10953 |
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