Theorem ifbieq12d2

Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017) (Proof shortened by Wolf Lammen, 24-Jun-2021)

	Ref	Expression
Hypotheses	ifbieq12d2.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	ifbieq12d2.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
	ifbieq12d2.3	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 = 𝐷 )
Assertion	ifbieq12d2	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ifbieq12d2.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		ifbieq12d2.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
3		ifbieq12d2.3	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 = 𝐷 )
4	2 3	ifeq12da	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )
5	1	ifbid	⊢ ( 𝜑 → if ( 𝜓 , 𝐶 , 𝐷 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )
6	4 5	eqtrd	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) )