Theorem brif2 41703

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 4531	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq2d 5155	. 2 ⊢ (𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐴))
3		iffalse 4534	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq2d 5155	. 2 ⊢ (¬ 𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐵))
5	2, 4	casesifp 1076	1 ⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 205  if-wif 1061  ifcif 4525   class class class wbr 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144

This theorem is referenced by:  prjspner01  42040