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Theorem brif2 41350
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
Assertion
Ref Expression
brif2 (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))

Proof of Theorem brif2
StepHypRef Expression
1 iftrue 4535 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21breq2d 5161 . 2 (𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐴))
3 iffalse 4538 . . 3 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43breq2d 5161 . 2 𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐵))
52, 4casesifp 1075 1 (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  if-wif 1059  ifcif 4529   class class class wbr 5149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150
This theorem is referenced by:  prjspner01  41671
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