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Theorem ineqri 3264
Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-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 3254 . . 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 183 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  C
)
43eqriv 2134 1  |-  ( A  i^i  B )  =  C
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    i^i cin 3065
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072
This theorem is referenced by:  inidm  3280  inass  3281  indi  3318  inab  3339  in0  3392  pwin  4199  dmres  4835
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