Theorem brif1 7511

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 4498	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq1d 5124	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐴𝑅𝐶))
3		iffalse 4501	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq1d 5124	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ 𝐵𝑅𝐶))
5	2, 4	casesifp 1092	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1076  ifcif 4492   class class class wbr 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115

This theorem is referenced by:  psdmul  22312  psdmvr  22315