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Mirrors > Home > MPE Home > Th. List > Mathboxes > brif1 | Structured version Visualization version GIF version |
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
brif1 | ⊢ (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4429 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | breq1d 5046 | . 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐴𝑅𝐶)) |
3 | iffalse 4432 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
4 | 3 | breq1d 5046 | . 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
5 | 2, 4 | casesifp 1074 | 1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 if-wif 1058 ifcif 4423 class class class wbr 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3an 1086 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-v 3411 df-un 3865 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-br 5037 |
This theorem is referenced by: (None) |
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