Theorem eqneltrrd

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) (Proof shortened by Wolf Lammen, 13-Nov-2019)

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

Proof

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