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Theorem difeqri 3372
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 3238 . . 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 240 . 2  |-  ( x  e.  ( A  \  B )  <->  x  e.  C )
43eqriv 2355 1  |-  ( A 
\  B )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225
This theorem is referenced by:  difdif  3378  ddif  3384  dfss4  3479  difin  3482  difab  3513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-dif 3231
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