Theorem brif1 7509

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 4511	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq1d 5134	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐴𝑅𝐶))
3		iffalse 4514	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq1d 5134	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐵𝑅𝐶))
5	2, 4	casesifp 1077	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 206  if-wif 1062  ifcif 4505   class class class wbr 5124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125

This theorem is referenced by:  psdmul  22109  psdmvr  22112