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Theorem ineqri 3375
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 3371 . . 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 240 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  C
)
43eqriv 2293 1  |-  ( A  i^i  B )  =  C
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164
This theorem is referenced by:  inidm  3391  inass  3392  dfin2  3418  indi  3428  inab  3449  in0  3493  pwin  4313  dmres  4992  dfres3  24187  inixp  26502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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