Theorem ineqri 3342

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 3332	. . 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 184	. 2 |- ( x e. ( A i^i B ) <-> x e. C )
4	3	eqriv 2185	1 |- ( A i^i B ) = C

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2159   i^i cin 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-in 3149

This theorem is referenced by:  inidm  3358  inass  3359  indi  3396  inab  3417  in0  3471  pwin  4296  dmres  4942