Theorem brif2 42241

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

Step	Hyp	Ref	Expression
1		iftrue 4536	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq2d 5159	. 2 ⊢ (𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐴))
3		iffalse 4539	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq2d 5159	. 2 ⊢ (¬ 𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐵))
5	2, 4	casesifp 1077	1 ⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206  if-wif 1062  ifcif 4530   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 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148

This theorem is referenced by:  prjspner01  42611