Theorem ineqri 3194

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 3184	. . 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 2086	1 |- ( A i^i B ) = C

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   i^i cin 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006

This theorem is referenced by:  inidm  3210  inass  3211  indi  3247  inab  3268  in0  3321  pwin  4118  dmres  4747