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Theorem iinin2 4036
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4020 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin2 (Ax A (BC) = (Bx 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.
StepHypRef Expression
1 r19.28zv 3645 . . . 4 (A → (x A (y B y C) ↔ (y B x A y C)))
2 elin 3219 . . . . 5 (y (BC) ↔ (y B y C))
32ralbii 2638 . . . 4 (x A y (BC) ↔ x A (y B y C))
4 vex 2862 . . . . . 6 y V
5 eliin 3974 . . . . . 6 (y V → (y x A Cx A y C))
64, 5ax-mp 5 . . . . 5 (y x A Cx A y C)
76anbi2i 675 . . . 4 ((y B y x A C) ↔ (y B x A y C))
81, 3, 73bitr4g 279 . . 3 (A → (x A y (BC) ↔ (y B y x A C)))
9 eliin 3974 . . . 4 (y V → (y x A (BC) ↔ x A y (BC)))
104, 9ax-mp 5 . . 3 (y x A (BC) ↔ x A y (BC))
11 elin 3219 . . 3 (y (Bx A C) ↔ (y B y x A C))
128, 10, 113bitr4g 279 . 2 (A → (y x A (BC) ↔ y (Bx A C)))
1312eqrdv 2351 1 (Ax A (BC) = (Bx A C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  Vcvv 2859   ∩ cin 3208  ∅c0 3550  ∩ciin 3970 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-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iin 3972 This theorem is referenced by:  iinin1  4037
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