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Theorem difeqri 3197
Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypothesis
Ref Expression
difeqri.1  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
)
Assertion
Ref Expression
difeqri  |-  ( A 
\  B )  =  C
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 3081 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 difeqri.1 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
)
31, 2bitri 183 . 2  |-  ( x  e.  ( A  \  B )  <->  x  e.  C )
43eqriv 2137 1  |-  ( A 
\  B )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481    \ cdif 3069
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-v 2689  df-dif 3074
This theorem is referenced by:  difdif  3202  ddifnel  3208  difab  3346
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