Theorem ifcld

Description: Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018)

	Ref	Expression
Hypotheses	ifcld.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
	ifcld.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
Assertion	ifcld	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ifcld.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2		ifcld.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
3		ifcl	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )