Theorem ifcl

Description: Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005)

	Ref	Expression
Assertion	ifcl	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝐴 ∈ 𝐶 ↔ if ( 𝜑 , 𝐴 , 𝐵 ) ∈ 𝐶 ) )
2		eleq1	⊢ ( 𝐵 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝐵 ∈ 𝐶 ↔ if ( 𝜑 , 𝐴 , 𝐵 ) ∈ 𝐶 ) )
3	1 2	ifboth	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ 𝐶 )