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Mirrors > Home > ILE Home > Th. List > ifeq12d | GIF version |
Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Ref | Expression |
---|---|
ifeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
ifeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
ifeq12d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | ifeq1d 3412 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | ifeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | ifeq2d 3413 | . 2 ⊢ (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
5 | 2, 4 | eqtrd 2121 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ifcif 3397 |
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 2622 df-un 3004 df-if 3398 |
This theorem is referenced by: ifbieq12d 3421 exp3val 10011 eucalgval 11368 |
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