Theorem dfin5 3181

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 3180	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2495	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2231	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   {crab 2490   i^i cin 3173

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-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-rab 2495  df-in 3180

This theorem is referenced by:  nfin  3387  rabbi2dva  3389  ssfidc  7060  suprzubdc  10416  nninfdcex  10417  nnmindc  12470  nnminle  12471  znnen  12884  bj-inex  16042  2omap  16132  pw1map  16134