Theorem ifeq12

Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004)

	Ref	Expression
Assertion	ifeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → if ( 𝜑 , 𝐴 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq1	⊢ ( 𝐴 = 𝐵 → if ( 𝜑 , 𝐴 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐶 ) )
2		ifeq2	⊢ ( 𝐶 = 𝐷 → if ( 𝜑 , 𝐵 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐷 ) )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → if ( 𝜑 , 𝐴 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐷 ) )