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	|- ( 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	eqeq1d	|- ( ph -> ( A = C <-> B = C ) )
4	2	eqeq2d	|- ( ps -> ( B = C <-> B = D ) )
5	3 4	sylan9bb	|- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )