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Theorem difeqri 3469
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 3332 . . 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 242 . 2  |-  ( x  e.  ( A  \  B )  <->  x  e.  C )
43eqriv 2435 1  |-  ( A 
\  B )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319
This theorem is referenced by:  difdif  3475  ddif  3481  dfss4  3577  difin  3580  difab  3612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325
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