Theorem eleqtrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		eleqtrd.1	|- ( ph -> A e. B )
2		eleqtrd.2	|- ( ph -> B = C )
3	2	eleq2d	|- ( ph -> ( A e. B <-> A e. C ) )
4	1 3	mpbid	|- ( ph -> A e. C )