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Theorem iinin2m 3753
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 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 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 3342 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝑦𝐶)))
2 elin 3154 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32ralbii 2347 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 vex 2577 . . . . . 6 𝑦 ∈ V
5 eliin 3690 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
64, 5ax-mp 7 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
76anbi2i 438 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝑦𝐶))
81, 3, 73bitr4g 216 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶)))
9 eliin 3690 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
104, 9ax-mp 7 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
11 elin 3154 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
128, 10, 113bitr4g 216 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1312eqrdv 2054 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wral 2323  Vcvv 2574  cin 2944   ciin 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-iin 3688
This theorem is referenced by:  iinin1m  3754
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