Theorem dfin5 3004

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 3003	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2368	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2111	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 102   = wceq 1289   e. wcel 1438   {cab 2074   {crab 2363   i^i cin 2996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-rab 2368  df-in 3003

This theorem is referenced by:  nfin  3204  rabbi2dva  3206  ssfidc  6623  znnen  11293  bj-inex  11444