Theorem brif1 7495

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

Step	Hyp	Ref	Expression
1		iftrue 4488	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq1d 5112	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐴𝑅𝐶))
3		iffalse 4491	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq1d 5112	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐵𝑅𝐶))
5	2, 4	casesifp 1090	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 208  if-wif 1074  ifcif 4482   class class class wbr 5102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1075  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103

This theorem is referenced by:  psdmul  22233  psdmvr  22236