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Theorem ineqri 3633
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 3626 . . 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 2398 1  |-  ( A  i^i  B )  =  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    i^i cin 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-in 3421
This theorem is referenced by:  inidm  3648  inass  3649  dfin2  3686  indi  3696  inab  3718  in0  3765  pwin  4727  dmres  5114  dfres3  29972  inixp  31501
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