Theorem eqeqan12rd

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994)

	Ref	Expression
Hypotheses	eqeqan12rd.1	$$⊢\phi \to A=B$$
	eqeqan12rd.2	$$⊢\psi \to C=D$$
Assertion	eqeqan12rd	$$⊢\left(\psi \wedge \phi \right)\to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeqan12rd.1	$$⊢\phi \to A=B$$
2		eqeqan12rd.2	$$⊢\psi \to C=D$$
3	1 2	eqeqan12d	$$⊢\left(\phi \wedge \psi \right)\to \left(A=C\leftrightarrow B=D\right)$$
4	3	ancoms	$$⊢\left(\psi \wedge \phi \right)\to \left(A=C\leftrightarrow B=D\right)$$