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	|- ( ph -> A = B )
	neleq12d.2	|- ( ph -> C = D )
Assertion	neleq12d	|- ( ph -> ( A e/ C <-> B e/ D ) )

Proof

Step	Hyp	Ref	Expression
1		neleq12d.1	|- ( ph -> A = B )
2		neleq12d.2	|- ( ph -> C = D )
3	1 2	eleq12d	|- ( ph -> ( A e. C <-> B e. D ) )
4	3	notbid	|- ( ph -> ( -. A e. C <-> -. B e. D ) )
5		df-nel	|- ( A e/ C <-> -. A e. C )
6		df-nel	|- ( B e/ D <-> -. B e. D )
7	4 5 6	3bitr4g	|- ( ph -> ( A e/ C <-> B e/ D ) )