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Mirrors > Home > MPE Home > Th. List > Mathboxes > brif2 | Structured version Visualization version GIF version |
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
brif2 | ⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4533 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | breq2d 5159 | . 2 ⊢ (𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐴)) |
3 | iffalse 4536 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
4 | 3 | breq2d 5159 | . 2 ⊢ (¬ 𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐵)) |
5 | 2, 4 | casesifp 1078 | 1 ⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 if-wif 1062 ifcif 4527 class class class wbr 5147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 |
This theorem is referenced by: prjspner01 41311 |
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