Theorem dfint2 4477

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 4476	. 2 |- |^| A = { x | A. y ( y e. A -> x e. y ) }
2		df-ral 2917	. . 3 |- ( A. y e. A x e. y <-> A. y ( y e. A -> x e. y ) )
3	2	abbii 2739	. 2 |- { x | A. y e. A x e. y } = { x | A. y ( y e. A -> x e. y ) }
4	1, 3	eqtr4i 2647	1 |- |^| A = { x | A. y e. A x e. y }

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-int 4476

This theorem is referenced by:  inteq  4478  elintg  4483  nfint  4486  intss  4498  intiin  4574  dfint3  32059