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

Proof

Step	Hyp	Ref	Expression
1		eqneltrd.1	$$⊢\phi \to A=B$$
2		eqneltrd.2	$$⊢\phi \to \neg B\in C$$
3	1	eleq1d	$$⊢\phi \to \left(A\in C\leftrightarrow B\in C\right)$$
4	2 3	mtbird	$$⊢\phi \to \neg A\in C$$