Theorem iunin2 2030

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 2026 to recover Enderton's theorem.

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem iunin2

Step	Hyp	Ref	Expression
1		r19.42v 1303	. . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ y ∈ C) ↔ (y ∈ B ∧ ∃x ∈ A y ∈ C))
2		elin 1635	. . . . 5 ⊢ (y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ C))
3	2	birex 1224	. . . 4 ⊢ (∃x ∈ A y ∈ (B ∩ C) ↔ ∃x ∈ A (y ∈ B ∧ y ∈ C))
4		eliun 1998	. . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
5	4	anbi2i 367	. . . 4 ⊢ ((y ∈ B ∧ y ∈ ∪x ∈ A C) ↔ (y ∈ B ∧ ∃x ∈ A y ∈ C))
6	1, 3, 5	3bitr4 158	. . 3 ⊢ (∃x ∈ A y ∈ (B ∩ C) ↔ (y ∈ B ∧ y ∈ ∪x ∈ A C))
7		eliun 1998	. . 3 ⊢ (y ∈ ∪x ∈ A (B ∩ C) ↔ ∃x ∈ A y ∈ (B ∩ C))
8		elin 1635	. . 3 ⊢ (y ∈ (B ∩ ∪x ∈ A C) ↔ (y ∈ B ∧ y ∈ ∪x ∈ A C))
9	6, 7, 8	3bitr4 158	. 2 ⊢ (y ∈ ∪x ∈ A (B ∩ C) ↔ y ∈ (B ∩ ∪x ∈ A C))
10	9	cleqri 1101	1 ⊢ ∪x ∈ A (B ∩ C) = (B ∩ ∪x ∈ A C)

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∩ cin 1486  ∪ciun 1994

This theorem is referenced by:  kmlem10 3589

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-in 1491  df-iun 1996

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