Theorem ineqri 3269

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 3259	. . 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 2136	1 |- ( A i^i B ) = C

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   i^i cin 3070

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077

This theorem is referenced by:  inidm  3285  inass  3286  indi  3323  inab  3344  in0  3397  pwin  4204  dmres  4840