Theorem dfint2 3925

Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
dfint2	|- |^| A = { x | A. y e. A x e. y }

Distinct variable group:   x, y, A

Proof of Theorem dfint2

Step	Hyp	Ref	Expression
1		df-int 3924	. 2 |- |^| A = { x | A. y ( y e. A -> x e. y ) }
2		df-ral 2513	. . 3 |- ( A. y e. A x e. y <-> A. y ( y e. A -> x e. y ) )
3	2	abbii 2345	. 2 |- { x | A. y e. A x e. y } = { x | A. y ( y e. A -> x e. y ) }
4	1, 3	eqtr4i 2253	1 |- |^| A = { x | A. y e. A x e. y }

Syntax hints:   -> wi 4   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   |^|cint 3923

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-ral 2513  df-int 3924

This theorem is referenced by:  inteq  3926  nfint  3933  intiin  4020