Theorem dfin5 3122

Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Assertion

Ref	Expression
dfin5	|- ( A i^i B ) = { x e. A | x e. B }

Distinct variable groups:   x, A   x, B

Proof of Theorem dfin5

Step	Hyp	Ref	Expression
1		df-in 3121	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2452	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2189	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2447   i^i cin 3114

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-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-rab 2452  df-in 3121

This theorem is referenced by:  nfin  3327  rabbi2dva  3329  ssfidc  6896  suprzubdc  11881  nninfdcex  11882  nnmindc  11963  nnminle  11964  znnen  12327  bj-inex  13749