Theorem nelrdva 2987

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 2208	. 2 |- ( ( ph /\ B e. A ) -> B = B )
2		eleq1 2270	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
3	2	anbi2d 464	. . . . . 6 |- ( x = B -> ( ( ph /\ x e. A ) <-> ( ph /\ B e. A ) ) )
4		neeq1 2391	. . . . . 6 |- ( x = B -> ( x =/= B <-> B =/= B ) )
5	3, 4	imbi12d 234	. . . . 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 2838	. . . 4 |- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7 581	. . 3 |- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd 2399	. 2 |- ( ( ph /\ B e. A ) -> -. B = B )
10	1, 9	pm2.65da 663	1 |- ( ph -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   =/= wne 2378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778

This theorem is referenced by: (None)