Theorem dfint2 1967

Description: Alternate definition of class intersection.

Assertion

Ref	Expression
dfint2	⊢ ∩A = {x∣∀y ∈ A x ∈ y}

Distinct variable group(s):   x,y,A

Proof of Theorem dfint2

Step	Hyp	Ref	Expression
1		df-int 1966	. 2 ⊢ ∩A = {x∣∀y(y ∈ A → x ∈ y)}
2		df-ral 1205	. . 3 ⊢ (∀y ∈ A x ∈ y ↔ ∀y(y ∈ A → x ∈ y))
3	2	biabi 1181	. 2 ⊢ {x∣∀y ∈ A x ∈ y} = {x∣∀y(y ∈ A → x ∈ y)}
4	1, 3	eqtr4 1122	1 ⊢ ∩A = {x∣∀y ∈ A x ∈ y}

Syntax hints:   → wi 2  ∀wal 672   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∩cint 1965

This theorem is referenced by:  inteq 1968  intiin 2027

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-int 1966

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