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Theorem nelrdva 3166
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
StepHypRef Expression
1 eqidd 2442 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  =  B )
2 eleq1 2501 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
32anbi2d 703 . . . . . 6  |-  ( x  =  B  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  B  e.  A ) ) )
4 neeq1 2614 . . . . . 6  |-  ( x  =  B  ->  (
x  =/=  B  <->  B  =/=  B ) )
53, 4imbi12d 320 . . . . 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 )
75, 6vtoclg 3028 . . . 4  |-  ( B  e.  A  ->  (
( ph  /\  B  e.  A )  ->  B  =/=  B ) )
87anabsi7 815 . . 3  |-  ( (
ph  /\  B  e.  A )  ->  B  =/=  B )
98neneqd 2622 . 2  |-  ( (
ph  /\  B  e.  A )  ->  -.  B  =  B )
101, 9pm2.65da 576 1  |-  ( ph  ->  -.  B  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-ne 2606  df-v 2972
This theorem is referenced by:  ustfilxp  19785  metustfbasOLD  20138  metustfbas  20139
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