Theorem ifcld

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

	Ref	Expression
Hypotheses	ifcld.a	|- ( ph -> A e. C )
	ifcld.b	|- ( ph -> B e. C )
Assertion	ifcld	|- ( ph -> if ( ps , A , B ) e. C )

Proof

Step	Hyp	Ref	Expression
1		ifcld.a	|- ( ph -> A e. C )
2		ifcld.b	|- ( ph -> B e. C )
3		ifcl	|- ( ( A e. C /\ B e. C ) -> if ( ps , A , B ) e. C )
4	1 2 3	syl2anc	|- ( ph -> if ( ps , A , B ) e. C )