Theorem eqeq12d

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Oct-2024)

	Ref	Expression
Hypotheses	eqeq12d.1	$$⊢\phi \to A=B$$
	eqeq12d.2	$$⊢\phi \to C=D$$
Assertion	eqeq12d	$$⊢\phi \to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq12d.1	$$⊢\phi \to A=B$$
2		eqeq12d.2	$$⊢\phi \to C=D$$
3	1 2	eqeqan12d	$$⊢\left(\phi \wedge \phi \right)\to \left(A=C\leftrightarrow B=D\right)$$
4	3	anidms	$$⊢\phi \to \left(A=C\leftrightarrow B=D\right)$$