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Mirrors > Home > ILE Home > Th. List > ifbieq12i | GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Ref | Expression |
---|---|
ifbieq12i.1 | ⊢ (𝜑 ↔ 𝜓) |
ifbieq12i.2 | ⊢ 𝐴 = 𝐶 |
ifbieq12i.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
ifbieq12i | ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq12i.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | ifeq1 3472 | . . 3 ⊢ (𝐴 = 𝐶 → if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵) |
4 | ifbieq12i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
5 | ifbieq12i.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
6 | 4, 5 | ifbieq2i 3490 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷) |
7 | 3, 6 | eqtri 2158 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ifcif 3469 |
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 603 ax-in2 604 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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-un 3070 df-if 3470 |
This theorem is referenced by: (None) |
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