Theorem cdeqel

Description: Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypotheses	cdeqeq.1	$$⊢CondEq\left(x=y\to A=B\right)$$
	cdeqeq.2	$$⊢CondEq\left(x=y\to C=D\right)$$
Assertion	cdeqel	$$⊢CondEq\left(x=y\to \left(A\in C\leftrightarrow B\in D\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdeqeq.1	$$⊢CondEq\left(x=y\to A=B\right)$$
2		cdeqeq.2	$$⊢CondEq\left(x=y\to C=D\right)$$
3	1	cdeqri	$$⊢x=y\to A=B$$
4	2	cdeqri	$$⊢x=y\to C=D$$
5	3 4	eleq12d	$$⊢x=y\to \left(A\in C\leftrightarrow B\in D\right)$$
6	5	cdeqi	$$⊢CondEq\left(x=y\to \left(A\in C\leftrightarrow B\in D\right)\right)$$