Theorem eqeq12

Description: Equality relationship among four classes. (Contributed by NM, 3-Aug-1994) (Proof shortened by Wolf Lammen, 23-Oct-2024)

	Ref	Expression
Assertion	eqeq12	$$⊢\left(A=B\wedge C=D\right)\to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢A=B\to A=B$$
2		id	$$⊢C=D\to C=D$$
3	1 2	eqeqan12d	$$⊢\left(A=B\wedge C=D\right)\to \left(A=C\leftrightarrow B=D\right)$$