Theorem dfint2 2530

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 2529	. 2 |- |^|A = {x | A.y(y e. A -> x e. y)}
2		df-ral 1646	. . 3 |- (A.y e. A x e. y <-> A.y(y e. A -> x e. y))
3	2	abbii 1572	. 2 |- {x | A.y e. A x e. y} = {x | A.y(y e. A -> x e. y)}
4	1, 3	eqtr4 1495	1 |- |^|A = {x | A.y e. A x e. y}

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642  |^|cint 2528

This theorem is referenced by:  inteq 2531  intiin 2597

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-int 2529

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