Theorem nelrdva

Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017)

	Ref	Expression
Hypothesis	nelrdva.1	|- ( ( ph /\ x e. A ) -> x =/= B )
Assertion	nelrdva	|- ( ph -> -. B e. A )

Proof

Step	Hyp	Ref	Expression
1		nelrdva.1	|- ( ( ph /\ x e. A ) -> x =/= B )
2		eqidd	|- ( ( ph /\ B e. A ) -> B = B )
3		eleq1	|- ( x = B -> ( x e. A <-> B e. A ) )
4	3	anbi2d	|- ( x = B -> ( ( ph /\ x e. A ) <-> ( ph /\ B e. A ) ) )
5		neeq1	|- ( x = B -> ( x =/= B <-> B =/= B ) )
6	4 5	imbi12d	|- ( x = B -> ( ( ( ph /\ x e. A ) -> x =/= B ) <-> ( ( ph /\ B e. A ) -> B =/= B ) ) )
7	6 1	vtoclg	|- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7	|- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd	|- ( ( ph /\ B e. A ) -> -. B = B )
10	2 9	pm2.65da	|- ( ph -> -. B e. A )