Theorem brif12

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

	Ref	Expression
Assertion	brif12	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 if ( 𝜑 , 𝐶 , 𝐷 ) ↔ if- ( 𝜑 , 𝐴 𝑅 𝐶 , 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
2		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐶 , 𝐷 ) = 𝐶 )
3	1 2	breq12d	⊢ ( 𝜑 → ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 if ( 𝜑 , 𝐶 , 𝐷 ) ↔ 𝐴 𝑅 𝐶 ) )
4		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
5		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐶 , 𝐷 ) = 𝐷 )
6	4 5	breq12d	⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 if ( 𝜑 , 𝐶 , 𝐷 ) ↔ 𝐵 𝑅 𝐷 ) )
7	3 6	casesifp	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) 𝑅 if ( 𝜑 , 𝐶 , 𝐷 ) ↔ if- ( 𝜑 , 𝐴 𝑅 𝐶 , 𝐵 𝑅 𝐷 ) )