Theorem iinin2 4037

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4021 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin2	⊢ (A ≠ ∅ → ∩x ∈ A (B ∩ C) = (B ∩ ∩x ∈ A C))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iinin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 3646	. . . 4 ⊢ (A ≠ ∅ → (∀x ∈ A (y ∈ B ∧ y ∈ C) ↔ (y ∈ B ∧ ∀x ∈ A y ∈ C)))
2		elin 3220	. . . . 5 ⊢ (y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ C))
3	2	ralbii 2639	. . . 4 ⊢ (∀x ∈ A y ∈ (B ∩ C) ↔ ∀x ∈ A (y ∈ B ∧ y ∈ C))
4		vex 2863	. . . . . 6 ⊢ y ∈ V
5		eliin 3975	. . . . . 6 ⊢ (y ∈ V → (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C)
7	6	anbi2i 675	. . . 4 ⊢ ((y ∈ B ∧ y ∈ ∩x ∈ A C) ↔ (y ∈ B ∧ ∀x ∈ A y ∈ C))
8	1, 3, 7	3bitr4g 279	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ ∩x ∈ A C)))
9		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A (B ∩ C) ↔ ∀x ∈ A y ∈ (B ∩ C)))
10	4, 9	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A (B ∩ C) ↔ ∀x ∈ A y ∈ (B ∩ C))
11		elin 3220	. . 3 ⊢ (y ∈ (B ∩ ∩x ∈ A C) ↔ (y ∈ B ∧ y ∈ ∩x ∈ A C))
12	8, 10, 11	3bitr4g 279	. 2 ⊢ (A ≠ ∅ → (y ∈ ∩x ∈ A (B ∩ C) ↔ y ∈ (B ∩ ∩x ∈ A C)))
13	12	eqrdv 2351	1 ⊢ (A ≠ ∅ → ∩x ∈ A (B ∩ C) = (B ∩ ∩x ∈ A C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  Vcvv 2860   ∩ cin 3209  ∅c0 3551  ∩ciin 3971

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-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-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iin 3973

This theorem is referenced by:  iinin1  4038