Theorem brif2

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

	Ref	Expression
Assertion	brif2	⊢ ( 𝐶 𝑅 if ( 𝜑 , 𝐴 , 𝐵 ) ↔ if- ( 𝜑 , 𝐶 𝑅 𝐴 , 𝐶 𝑅 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
2	1	breq2d	⊢ ( 𝜑 → ( 𝐶 𝑅 if ( 𝜑 , 𝐴 , 𝐵 ) ↔ 𝐶 𝑅 𝐴 ) )
3		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
4	3	breq2d	⊢ ( ¬ 𝜑 → ( 𝐶 𝑅 if ( 𝜑 , 𝐴 , 𝐵 ) ↔ 𝐶 𝑅 𝐵 ) )
5	2 4	casesifp	⊢ ( 𝐶 𝑅 if ( 𝜑 , 𝐴 , 𝐵 ) ↔ if- ( 𝜑 , 𝐶 𝑅 𝐴 , 𝐶 𝑅 𝐵 ) )