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) (Proof shortened by Wolf Lammen, 20-Nov-2019)

	Ref	Expression
Hypotheses	eqeqan12d.1	|- ( ph -> A = B )
	eqeqan12d.2	|- ( ps -> C = D )
Assertion	eqeqan12d	|- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )

Proof

Step	Hyp	Ref	Expression
1		eqeqan12d.1	|- ( ph -> A = B )
2		eqeqan12d.2	|- ( ps -> C = D )
3	1	adantr	|- ( ( ph /\ ps ) -> A = B )
4	2	adantl	|- ( ( ph /\ ps ) -> C = D )
5	3 4	eqeq12d	|- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )