Theorem brif12 43355

Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)

Assertion

Ref	Expression
brif12	⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))

Proof of Theorem brif12

Step	Hyp	Ref	Expression
1		iftrue 4487	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		iftrue 4487	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
3	1, 2	breq12d 5115	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐴𝑅𝐶))
4		iffalse 4490	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
5		iffalse 4490	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
6	4, 5	breq12d 5115	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐵𝑅𝐷))
7	3, 6	casesifp 1094	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))

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 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  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 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by: (None)