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Theorem iindif2m 3850
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 3425 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶)))
2 eldif 3050 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32bicomi 131 . . . . 5 ((𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ 𝑦 ∈ (𝐵𝐶))
43ralbii 2418 . . . 4 (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
5 ralnex 2403 . . . . . 6 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∃𝑥𝐴 𝑦𝐶)
6 eliun 3787 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
75, 6xchbinxr 657 . . . . 5 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
87anbi2i 452 . . . 4 ((𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
91, 4, 83bitr3g 221 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶)))
10 vex 2663 . . . 4 𝑦 ∈ V
11 eliin 3788 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
1210, 11ax-mp 5 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 3050 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
149, 12, 133bitr4g 222 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1514eqrdv 2115 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  wral 2393  wrex 2394  Vcvv 2660  cdif 3038   ciun 3783   ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-iun 3785  df-iin 3786
This theorem is referenced by: (None)
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