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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 (BC) = (Bx 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.
StepHypRef Expression
1 r19.42v 2765 . . . 4 (x A (y B y C) ↔ (y B x A y C))
2 elin 3219 . . . . 5 (y (BC) ↔ (y B y C))
32rexbii 2639 . . . 4 (x A y (BC) ↔ x A (y B y C))
4 eliun 3973 . . . . 5 (y x A Cx A y C)
54anbi2i 675 . . . 4 ((y B y x A C) ↔ (y B x A y C))
61, 3, 53bitr4i 268 . . 3 (x A y (BC) ↔ (y B y x A C))
7 eliun 3973 . . 3 (y x A (BC) ↔ x A y (BC))
8 elin 3219 . . 3 (y (Bx A C) ↔ (y B y x A C))
96, 7, 83bitr4i 268 . 2 (y x A (BC) ↔ y (Bx A C))
109eqriv 2350 1 x A (BC) = (Bx A C)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208  ∪ciun 3969 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 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
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