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Theorem nelrdva 2944
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 2178 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  =  B )
2 eleq1 2240 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
32anbi2d 464 . . . . . 6  |-  ( x  =  B  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  B  e.  A ) ) )
4 neeq1 2360 . . . . . 6  |-  ( x  =  B  ->  (
x  =/=  B  <->  B  =/=  B ) )
53, 4imbi12d 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 )
75, 6vtoclg 2797 . . . 4  |-  ( B  e.  A  ->  (
( ph  /\  B  e.  A )  ->  B  =/=  B ) )
87anabsi7 581 . . 3  |-  ( (
ph  /\  B  e.  A )  ->  B  =/=  B )
98neneqd 2368 . 2  |-  ( (
ph  /\  B  e.  A )  ->  -.  B  =  B )
101, 9pm2.65da 661 1  |-  ( ph  ->  -.  B  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    =/= wne 2347
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739
This theorem is referenced by: (None)
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