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Theorem iindif2m 3753
Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iindif2m (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iindif2m
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3341 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶)))
2 eldif 2983 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32bicomi 130 . . . . 5 ((𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ 𝑦 ∈ (𝐵𝐶))
43ralbii 2373 . . . 4 (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
5 ralnex 2359 . . . . . 6 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∃𝑥𝐴 𝑦𝐶)
6 eliun 3690 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
75, 6xchbinxr 641 . . . . 5 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
87anbi2i 445 . . . 4 ((𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
91, 4, 83bitr3g 220 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶)))
10 vex 2605 . . . 4 𝑦 ∈ V
11 eliin 3691 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
1210, 11ax-mp 7 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 2983 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
149, 12, 133bitr4g 221 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1514eqrdv 2080 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wral 2349  wrex 2350  Vcvv 2602  cdif 2971   ciun 3686   ciin 3687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-iun 3688  df-iin 3689
This theorem is referenced by: (None)
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