Theorem ifcl

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

	Ref	Expression
Assertion	ifcl	$$⊢\left(A\in C\wedge B\in C\right)\to if\left(\phi ,A,B\right)\in C$$

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢A=if\left(\phi ,A,B\right)\to \left(A\in C\leftrightarrow if\left(\phi ,A,B\right)\in C\right)$$
2		eleq1	$$⊢B=if\left(\phi ,A,B\right)\to \left(B\in C\leftrightarrow if\left(\phi ,A,B\right)\in C\right)$$
3	1 2	ifboth	$$⊢\left(A\in C\wedge B\in C\right)\to if\left(\phi ,A,B\right)\in C$$