Theorem dfin5 3109

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 3108	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2444	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2181	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 103   = wceq 1335   e. wcel 2128   {cab 2143   {crab 2439   i^i cin 3101

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-rab 2444  df-in 3108

This theorem is referenced by:  nfin  3313  rabbi2dva  3315  ssfidc  6876  nninfdcex  11833  znnen  12114  nnmindc  12164  nnminle  12165  bj-inex  13469