Theorem dfin5 3160

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 3159	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2481	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2217	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   i^i cin 3152

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-rab 2481  df-in 3159

This theorem is referenced by:  nfin  3365  rabbi2dva  3367  ssfidc  6991  suprzubdc  12089  nninfdcex  12090  nnmindc  12171  nnminle  12172  znnen  12555  bj-inex  15399