Theorem nelrdva 3417

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 2623	. 2 |- ( ( ph /\ B e. A ) -> B = B )
2		eleq1 2689	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
3	2	anbi2d 740	. . . . . 6 |- ( x = B -> ( ( ph /\ x e. A ) <-> ( ph /\ B e. A ) ) )
4		neeq1 2856	. . . . . 6 |- ( x = B -> ( x =/= B <-> B =/= B ) )
5	3, 4	imbi12d 334	. . . . 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 3266	. . . 4 |- ( B e. A -> ( ( ph /\ B e. A ) -> B =/= B ) )
8	7	anabsi7 860	. . 3 |- ( ( ph /\ B e. A ) -> B =/= B )
9	8	neneqd 2799	. 2 |- ( ( ph /\ B e. A ) -> -. B = B )
10	1, 9	pm2.65da 600	1 |- ( ph -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-v 3202

This theorem is referenced by:  ustfilxp  22016  metustfbas  22362  fourierdlem72  40395