Theorem dfint2 2535

Description: Alternate definition of class intersection.

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 2534	. 2 |- |^|A = {x | A.y(y e. A -> x e. y)}
2		df-ral 1649	. . 3 |- (A.y e. A x e. y <-> A.y(y e. A -> x e. y))
3	2	abbii 1575	. 2 |- {x | A.y e. A x e. y} = {x | A.y(y e. A -> x e. y)}
4	1, 3	eqtr4 1498	1 |- |^|A = {x | A.y e. A x e. y}

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  |^|cint 2533

This theorem is referenced by:  inteq 2536  intiin 2602

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-int 2534

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