Theorem nelrdva 2891

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

Hypothesis

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

Assertion

Ref	Expression
nelrdva	|- ( ph -> -. B e. A )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem nelrdva

Step	Hyp	Ref	Expression
1		eqidd 2140	. 2 |- ( ( ph /\ B e. A ) -> B = B )
2		eleq1 2202	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
3	2	anbi2d 459	. . . . . 6 |- ( x = B -> ( ( ph /\ x e. A ) <-> ( ph /\ B e. A ) ) )
4		neeq1 2321	. . . . . 6 |- ( x = B -> ( x =/= B <-> B =/= B ) )
5	3, 4	imbi12d 233	. . . . 5 |- ( x = B -> ( ( ( ph /\ x e. A ) -> x =/= B ) <-> ( ( ph /\ B e. A ) -> B =/= B ) ) )
6		nelrdva.1	. . . . 5 |- ( ( ph /\ x e. A ) -> x =/= B )
7	5, 6	vtoclg 2746	. . . 4 |- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7 570	. . 3 |- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd 2329	. 2 |- ( ( ph /\ B e. A ) -> -. B = B )
10	1, 9	pm2.65da 650	1 |- ( ph -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688

This theorem is referenced by: (None)