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)

	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		eqeq12	$$⊢\left(A=B\wedge C=D\right)\to \left(A=C\leftrightarrow B=D\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A=C\leftrightarrow B=D\right)$$