Theorem elin3 3448

Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypothesis

Ref	Expression
elin3.x	⊢ X = ((B ∩ C) ∩ D)

Assertion

Ref	Expression
elin3	⊢ (A ∈ X ↔ (A ∈ B ∧ A ∈ C ∧ A ∈ D))

Proof of Theorem elin3

Step	Hyp	Ref	Expression
1		elin 3220	. . 3 ⊢ (A ∈ (B ∩ C) ↔ (A ∈ B ∧ A ∈ C))
2	1	anbi1i 676	. 2 ⊢ ((A ∈ (B ∩ C) ∧ A ∈ D) ↔ ((A ∈ B ∧ A ∈ C) ∧ A ∈ D))
3		elin3.x	. . 3 ⊢ X = ((B ∩ C) ∩ D)
4	3	elin2 3447	. 2 ⊢ (A ∈ X ↔ (A ∈ (B ∩ C) ∧ A ∈ D))
5		df-3an 936	. 2 ⊢ ((A ∈ B ∧ A ∈ C ∧ A ∈ D) ↔ ((A ∈ B ∧ A ∈ C) ∧ A ∈ D))
6	2, 4, 5	3bitr4i 268	1 ⊢ (A ∈ X ↔ (A ∈ B ∧ A ∈ C ∧ A ∈ D))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ∩ cin 3209

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-3an 936  df-nan 1288  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-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by: (None)