Theorem dfin5 3128

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 3127	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2457	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2194	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   i^i cin 3120

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-rab 2457  df-in 3127

This theorem is referenced by:  nfin  3333  rabbi2dva  3335  ssfidc  6912  suprzubdc  11907  nninfdcex  11908  nnmindc  11989  nnminle  11990  znnen  12353  bj-inex  13942