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Mirrors > Home > MPE Home > Th. List > Mathboxes > brif12 | Structured version Visualization version GIF version |
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
brif12 | ⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4462 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | iftrue 4462 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶) | |
3 | 1, 2 | breq12d 5083 | . 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐴𝑅𝐶)) |
4 | iffalse 4465 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
5 | iffalse 4465 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷) | |
6 | 4, 5 | breq12d 5083 | . 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐵𝑅𝐷)) |
7 | 3, 6 | casesifp 1075 | 1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 if-wif 1059 ifcif 4456 class class class wbr 5070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 |
This theorem is referenced by: (None) |
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