Theorem neleqtrrd

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 13-Nov-2019)

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

Proof

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