Theorem eleq12d

Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994)

	Ref	Expression
Hypotheses	eleq12d.1	|- ( ph -> A = B )
	eleq12d.2	|- ( ph -> C = D )
Assertion	eleq12d	|- ( ph -> ( A e. C <-> B e. D ) )

Proof

Step	Hyp	Ref	Expression
1		eleq12d.1	|- ( ph -> A = B )
2		eleq12d.2	|- ( ph -> C = D )
3	2	eleq2d	|- ( ph -> ( A e. C <-> A e. D ) )
4	1	eleq1d	|- ( ph -> ( A e. D <-> B e. D ) )
5	3 4	bitrd	|- ( ph -> ( A e. C <-> B e. D ) )