Theorem ineqri 3315

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

Step	Hyp	Ref	Expression
1		elin 3305	. . 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 )
3	1, 2	bitri 183	. 2 |- ( x e. ( A i^i B ) <-> x e. C )
4	3	eqriv 2162	1 |- ( A i^i B ) = C

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   i^i cin 3115

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  inidm  3331  inass  3332  indi  3369  inab  3390  in0  3443  pwin  4260  dmres  4905