MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brif1 Structured version   Visualization version   GIF version

Theorem brif1 7530
Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
Assertion
Ref Expression
brif1 (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))

Proof of Theorem brif1
StepHypRef Expression
1 iftrue 4537 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21breq1d 5158 . 2 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶𝐴𝑅𝐶))
3 iffalse 4540 . . 3 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43breq1d 5158 . 2 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶𝐵𝑅𝐶))
52, 4casesifp 1077 1 (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  if-wif 1062  ifcif 4531   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149
This theorem is referenced by:  psdmul  22188
  Copyright terms: Public domain W3C validator