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	$$⊢\phi \to \neg C\in B$$
	neleqtrrd.2	$$⊢\phi \to A=B$$
Assertion	neleqtrrd	$$⊢\phi \to \neg C\in A$$

Proof

Step	Hyp	Ref	Expression
1		neleqtrrd.1	$$⊢\phi \to \neg C\in B$$
2		neleqtrrd.2	$$⊢\phi \to A=B$$
3	2	eqcomd	$$⊢\phi \to B=A$$
4	1 3	neleqtrd	$$⊢\phi \to \neg C\in A$$