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	|- ( ph -> A = B )
	eqeq12d.2	|- ( ph -> C = D )
Assertion	eqeq12d	|- ( ph -> ( A = C <-> B = D ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq12d.1	|- ( ph -> A = B )
2		eqeq12d.2	|- ( ph -> C = D )
3	1 2	eqeqan12d	|- ( ( ph /\ ph ) -> ( A = C <-> B = D ) )
4	3	anidms	|- ( ph -> ( A = C <-> B = D ) )