Theorem brif1

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

	Ref	Expression
Assertion	brif1	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 𝐶 ↔ if- ( 𝜑 , 𝐴 𝑅 𝐶 , 𝐵 𝑅 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
2	1	breq1d	⊢ ( 𝜑 → ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 𝐶 ↔ 𝐴 𝑅 𝐶 ) )
3		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
4	3	breq1d	⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
5	2 4	casesifp	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 𝐶 ↔ if- ( 𝜑 , 𝐴 𝑅 𝐶 , 𝐵 𝑅 𝐶 ) )