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Theorem ineqri 3685
Description: Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
ineqri.1  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )
Assertion
Ref Expression
ineqri  |-  ( A  i^i  B )  =  C
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem ineqri
StepHypRef Expression
1 elin 3680 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 ineqri.1 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )
31, 2bitri 249 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  C
)
43eqriv 2456 1  |-  ( A  i^i  B )  =  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476
This theorem is referenced by:  inidm  3700  inass  3701  dfin2  3727  indi  3737  inab  3759  in0  3804  pwin  4777  dmres  5285  dfres3  28751  inixp  29809
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