Theorem ifbieq12d

Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Hypotheses	ifbieq12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	ifbieq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	ifbieq12d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	ifbieq12d	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ifbieq12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		ifbieq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		ifbieq12d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	1	ifbid	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐴 , 𝐵 ) )
5	2 3	ifeq12d	⊢ ( 𝜑 → if ( 𝜒 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )
6	4 5	eqtrd	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )