Theorem brif2 43198

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 4487	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	breq2d 5114	. 2 ⊢ (𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐴))
3		iffalse 4490	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	breq2d 5114	. 2 ⊢ (¬ 𝜑 → (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ 𝐶𝑅𝐵))
5	2, 4	casesifp 1094	1 ⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  if-wif 1078  ifcif 4481   class class class wbr 5102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by:  prjspner01  43575