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Theorem brif12 40126
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
Assertion
Ref Expression
brif12 (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))

Proof of Theorem brif12
StepHypRef Expression
1 iftrue 4462 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 iftrue 4462 . . 3 (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
31, 2breq12d 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)
64, 5breq12d 5083 . 2 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐵𝑅𝐷))
73, 6casesifp 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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