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Theorem iindif2 5081
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 5062 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iindif2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iindif2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 4501 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶)))
2 eldif 3959 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32bicomi 223 . . . . 5 ((𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ 𝑦 ∈ (𝐵𝐶))
43ralbii 3094 . . . 4 (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
5 ralnex 3073 . . . . . 6 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∃𝑥𝐴 𝑦𝐶)
6 eliun 5002 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
75, 6xchbinxr 335 . . . . 5 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
87anbi2i 624 . . . 4 ((𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
91, 4, 83bitr3g 313 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶)))
10 eliin 5003 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
1110elv 3481 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
12 eldif 3959 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
139, 11, 123bitr4g 314 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1413eqrdv 2731 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  cdif 3946  c0 4323   ciun 4998   ciin 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-nul 4324  df-iun 5000  df-iin 5001
This theorem is referenced by:  iinvdif  5084  iincld  22543  clsval2  22554  mretopd  22596  hauscmplem  22910  cmpfi  22912  sigapildsyslem  33159
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