Theorem eleq12

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999)

	Ref	Expression
Assertion	eleq12	$$⊢\left(A=B\wedge C=D\right)\to \left(A\in C\leftrightarrow B\in D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢A=B\to \left(A\in C\leftrightarrow B\in C\right)$$
2		eleq2	$$⊢C=D\to \left(B\in C\leftrightarrow B\in D\right)$$
3	1 2	sylan9bb	$$⊢\left(A=B\wedge C=D\right)\to \left(A\in C\leftrightarrow B\in D\right)$$