Theorem fvifeq

Description: Equality of function values with conditional arguments, see also fvif . (Contributed by Alexander van der Vekens, 21-May-2018)

	Ref	Expression
Assertion	fvifeq	⊢ ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) → ( 𝐹 ‘ 𝐴 ) = if ( 𝜑 , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ if ( 𝜑 , 𝐵 , 𝐶 ) ) )
2		fvif	⊢ ( 𝐹 ‘ if ( 𝜑 , 𝐵 , 𝐶 ) ) = if ( 𝜑 , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) )
3	1 2	eqtrdi	⊢ ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) → ( 𝐹 ‘ 𝐴 ) = if ( 𝜑 , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) ) )