Theorem ifeq12d

Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015)

	Ref	Expression
Hypotheses	ifeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	ifeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	ifeq12d	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ifeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	ifeq1d	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )
4	2	ifeq2d	⊢ ( 𝜑 → if ( 𝜓 , 𝐵 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐷 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐷 ) )