Theorem nelrdva 2895

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 2141	. 2 |- ( ( ph /\ B e. A ) -> B = B )
2		eleq1 2203	. . . . . . 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 2322	. . . . . 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 2749	. . . 4 |- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7 571	. . 3 |- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd 2330	. 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 1332   e. wcel 1481   =/= wne 2309

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691

This theorem is referenced by: (None)