Theorem brif12 42317

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 4478	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		iftrue 4478	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
3	1, 2	breq12d 5102	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐴𝑅𝐶))
4		iffalse 4481	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
5		iffalse 4481	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
6	4, 5	breq12d 5102	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ 𝐵𝑅𝐷))
7	3, 6	casesifp 1077	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 206  if-wif 1062  ifcif 4472   class class class wbr 5089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090

This theorem is referenced by: (None)