Theorem elintg 3935

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Assertion

Ref	Expression
elintg	⊢ (A ∈ V → (A ∈ ∩B ↔ ∀x ∈ B A ∈ x))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   V(x)

Proof of Theorem elintg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (y = A → (y ∈ ∩B ↔ A ∈ ∩B))
2		eleq1 2413	. . 3 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
3	2	ralbidv 2635	. 2 ⊢ (y = A → (∀x ∈ B y ∈ x ↔ ∀x ∈ B A ∈ x))
4		vex 2863	. . 3 ⊢ y ∈ V
5	4	elint2 3934	. 2 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x)
6	1, 3, 5	vtoclbg 2916	1 ⊢ (A ∈ V → (A ∈ ∩B ↔ ∀x ∈ B A ∈ x))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-int 3928

This theorem is referenced by:  elinti  3936  elrint  3968