Theorem eqeqan12dALT

Description: Alternate proof of eqeqan12d . This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eqeqan12dALT.1	$$⊢\phi \to A=B$$
	eqeqan12dALT.2	$$⊢\psi \to C=D$$
Assertion	eqeqan12dALT	$$⊢\left(\phi \wedge \psi \right)\to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeqan12dALT.1	$$⊢\phi \to A=B$$
2		eqeqan12dALT.2	$$⊢\psi \to C=D$$
3		eqeq12	$$⊢\left(A=B\wedge C=D\right)\to \left(A=C\leftrightarrow B=D\right)$$
4	1 2 3	syl2an	$$⊢\left(\phi \wedge \psi \right)\to \left(A=C\leftrightarrow B=D\right)$$