Theorem nelrdva 2933

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 2166	. 2 |- ( ( ph /\ B e. A ) -> B = B )
2		eleq1 2229	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
3	2	anbi2d 460	. . . . . 6 |- ( x = B -> ( ( ph /\ x e. A ) <-> ( ph /\ B e. A ) ) )
4		neeq1 2349	. . . . . 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 2786	. . . 4 |- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7 571	. . 3 |- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd 2357	. 2 |- ( ( ph /\ B e. A ) -> -. B = B )
10	1, 9	pm2.65da 651	1 |- ( ph -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728

This theorem is referenced by: (None)