Theorem ifcld

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

	Ref	Expression
Hypotheses	ifcld.a	$$⊢\phi \to A\in C$$
	ifcld.b	$$⊢\phi \to B\in C$$
Assertion	ifcld	$$⊢\phi \to if\left(\psi ,A,B\right)\in C$$

Proof

Step	Hyp	Ref	Expression
1		ifcld.a	$$⊢\phi \to A\in C$$
2		ifcld.b	$$⊢\phi \to B\in C$$
3		ifcl	$$⊢\left(A\in C\wedge B\in C\right)\to if\left(\psi ,A,B\right)\in C$$
4	1 2 3	syl2anc	$$⊢\phi \to if\left(\psi ,A,B\right)\in C$$