Theorem eqeqan12d

Description: A useful inference for substituting definitions into an equality. See also eqeqan12dALT . (Contributed by NM, 9-Aug-1994) (Proof shortened by Andrew Salmon, 25-May-2011) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024)

	Ref	Expression
Hypotheses	eqeqan12d.1	$$⊢\phi \to A=B$$
	eqeqan12d.2	$$⊢\psi \to C=D$$
Assertion	eqeqan12d	$$⊢\left(\phi \wedge \psi \right)\to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeqan12d.1	$$⊢\phi \to A=B$$
2		eqeqan12d.2	$$⊢\psi \to C=D$$
3	1	eqeq1d	$$⊢\phi \to \left(A=C\leftrightarrow B=C\right)$$
4	2	eqeq2d	$$⊢\psi \to \left(B=C\leftrightarrow B=D\right)$$
5	3 4	sylan9bb	$$⊢\left(\phi \wedge \psi \right)\to \left(A=C\leftrightarrow B=D\right)$$