Theorem dfint2 3901

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 3900	. 2 |- |^| A = { x | A. y ( y e. A -> x e. y ) }
2		df-ral 2491	. . 3 |- ( A. y e. A x e. y <-> A. y ( y e. A -> x e. y ) )
3	2	abbii 2323	. 2 |- { x | A. y e. A x e. y } = { x | A. y ( y e. A -> x e. y ) }
4	1, 3	eqtr4i 2231	1 |- |^| A = { x | A. y e. A x e. y }

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   |^|cint 3899

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-ral 2491  df-int 3900

This theorem is referenced by:  inteq  3902  nfint  3909  intiin  3996