Theorem iunin2 4030

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4019 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
iunin2	⊢ ∪x ∈ A (B ∩ C) = (B ∩ ∪x ∈ A C)

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   C(x)

Proof of Theorem iunin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.42v 2765	. . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ y ∈ C) ↔ (y ∈ B ∧ ∃x ∈ A y ∈ C))
2		elin 3219	. . . . 5 ⊢ (y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ C))
3	2	rexbii 2639	. . . 4 ⊢ (∃x ∈ A y ∈ (B ∩ C) ↔ ∃x ∈ A (y ∈ B ∧ y ∈ C))
4		eliun 3973	. . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
5	4	anbi2i 675	. . . 4 ⊢ ((y ∈ B ∧ y ∈ ∪x ∈ A C) ↔ (y ∈ B ∧ ∃x ∈ A y ∈ C))
6	1, 3, 5	3bitr4i 268	. . 3 ⊢ (∃x ∈ A y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ ∪x ∈ A C))
7		eliun 3973	. . 3 ⊢ (y ∈ ∪x ∈ A (B ∩ C) ↔ ∃x ∈ A y ∈ (B ∩ C))
8		elin 3219	. . 3 ⊢ (y ∈ (B ∩ ∪x ∈ A C) ↔ (y ∈ B ∧ y ∈ ∪x ∈ A C))
9	6, 7, 8	3bitr4i 268	. 2 ⊢ (y ∈ ∪x ∈ A (B ∩ C) ↔ y ∈ (B ∩ ∪x ∈ A C))
10	9	eqriv 2350	1 ⊢ ∪x ∈ A (B ∩ C) = (B ∩ ∪x ∈ A C)

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208  ∪ciun 3969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-iun 3971

This theorem is referenced by:  iunin1  4031  2iunin  4034