Theorem dfint2 3929

Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Assertion

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

Distinct variable group:   x,y,A

Proof of Theorem dfint2

Step	Hyp	Ref	Expression
1		df-int 3928	. 2 ⊢ ∩A = {x ∣ ∀y(y ∈ A → x ∈ y)}
2		df-ral 2620	. . 3 ⊢ (∀y ∈ A x ∈ y ↔ ∀y(y ∈ A → x ∈ y))
3	2	abbii 2466	. 2 ⊢ {x ∣ ∀y ∈ A x ∈ y} = {x ∣ ∀y(y ∈ A → x ∈ y)}
4	1, 3	eqtr4i 2376	1 ⊢ ∩A = {x ∣ ∀y ∈ A x ∈ y}

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2620  df-int 3928

This theorem is referenced by:  inteq  3930  nfint  3937  intiin  4021