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Theorem iinin2m 3847
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 3421 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝑦𝐶)))
2 elin 3225 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32ralbii 2415 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 vex 2660 . . . . . 6 𝑦 ∈ V
5 eliin 3784 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
64, 5ax-mp 7 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
76anbi2i 450 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝑦𝐶))
81, 3, 73bitr4g 222 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶)))
9 eliin 3784 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
104, 9ax-mp 7 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
11 elin 3225 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
128, 10, 113bitr4g 222 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1312eqrdv 2113 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  wral 2390  Vcvv 2657  cin 3036   ciin 3780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-in 3043  df-iin 3782
This theorem is referenced by:  iinin1m  3848
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