Theorem eqneltrd

Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017)

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

Proof

Step	Hyp	Ref	Expression
1		eqneltrd.1	|- ( ph -> A = B )
2		eqneltrd.2	|- ( ph -> -. B e. C )
3	1	eleq1d	|- ( ph -> ( A e. C <-> B e. C ) )
4	2 3	mtbird	|- ( ph -> -. A e. C )