Theorem dfint2 3967

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 3966	. 2 |- |^| A = { x | A. y ( y e. A -> x e. y ) }
2		df-ral 2533	. . 3 |- ( A. y e. A x e. y <-> A. y ( y e. A -> x e. y ) )
3	2	abbii 2354	. 2 |- { x | A. y e. A x e. y } = { x | A. y ( y e. A -> x e. y ) }
4	1, 3	eqtr4i 2262	1 |- |^| A = { x | A. y e. A x e. y }

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   |^|cint 3965

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-ral 2533  df-int 3966

This theorem is referenced by:  inteq  3968  nfint  3975  intiin  4062