Theorem dfin5 3164

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 3163	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2484	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2220	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   {cab 2182   {crab 2479   i^i cin 3156

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-rab 2484  df-in 3163

This theorem is referenced by:  nfin  3369  rabbi2dva  3371  ssfidc  6998  suprzubdc  10326  nninfdcex  10327  nnmindc  12201  nnminle  12202  znnen  12615  bj-inex  15553