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Theorem iundif2 5074
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 5059 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iundif2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldif 3961 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 3094 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 3191 . . . 4 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
4 rexnal 3100 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∀𝑥𝐴 𝑦𝐶)
5 eliin 4996 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
65elv 3485 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
74, 6xchbinxr 335 . . . . 5 (∃𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
87anbi2i 623 . . . 4 ((𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
92, 3, 83bitri 297 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
10 eliun 4995 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
11 eldif 3961 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
129, 10, 113bitr4i 303 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1312eqriv 2734 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948   ciun 4991   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-iun 4993  df-iin 4994
This theorem is referenced by:  iuncld  23053  pnrmopn  23351  alexsublem  24052  bcth3  25365  iundifdifd  32574  iundifdif  32575
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