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Theorem iinin2m 3715
 Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin2m (x x A x A (B𝐶) = (B x A 𝐶))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem iinin2m
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3308 . . . 4 (x x A → (x A (y B y 𝐶) ↔ (y B x A y 𝐶)))
2 elin 3120 . . . . 5 (y (B𝐶) ↔ (y B y 𝐶))
32ralbii 2324 . . . 4 (x A y (B𝐶) ↔ x A (y B y 𝐶))
4 vex 2554 . . . . . 6 y V
5 eliin 3652 . . . . . 6 (y V → (y x A 𝐶x A y 𝐶))
64, 5ax-mp 7 . . . . 5 (y x A 𝐶x A y 𝐶)
76anbi2i 430 . . . 4 ((y B y x A 𝐶) ↔ (y B x A y 𝐶))
81, 3, 73bitr4g 212 . . 3 (x x A → (x A y (B𝐶) ↔ (y B y x A 𝐶)))
9 eliin 3652 . . . 4 (y V → (y x A (B𝐶) ↔ x A y (B𝐶)))
104, 9ax-mp 7 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
11 elin 3120 . . 3 (y (B x A 𝐶) ↔ (y B y x A 𝐶))
128, 10, 113bitr4g 212 . 2 (x x A → (y x A (B𝐶) ↔ y (B x A 𝐶)))
1312eqrdv 2035 1 (x x A x A (B𝐶) = (B x A 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ∩ cin 2910  ∩ ciin 3648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-iin 3650 This theorem is referenced by:  iinin1m  3716
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