Theorem neleq12d

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypotheses	neleq12d.1	$$⊢\phi \to A=B$$
	neleq12d.2	$$⊢\phi \to C=D$$
Assertion	neleq12d	$$⊢\phi \to \left(A\notin C\leftrightarrow B\notin D\right)$$

Proof

Step	Hyp	Ref	Expression
1		neleq12d.1	$$⊢\phi \to A=B$$
2		neleq12d.2	$$⊢\phi \to C=D$$
3	1 2	eleq12d	$$⊢\phi \to \left(A\in C\leftrightarrow B\in D\right)$$
4	3	notbid	$$⊢\phi \to \left(\neg A\in C\leftrightarrow \neg B\in D\right)$$
5		df-nel	$$⊢A\notin C\leftrightarrow \neg A\in C$$
6		df-nel	$$⊢B\notin D\leftrightarrow \neg B\in D$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(A\notin C\leftrightarrow B\notin D\right)$$