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Theorem ineqri 3544
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 3539 . . 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 2440 1  |-  ( A  i^i  B )  =  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-in 3335
This theorem is referenced by:  inidm  3559  inass  3560  dfin2  3586  indi  3596  inab  3618  in0  3663  pwin  4625  dmres  5131  dfres3  27569  inixp  28622
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