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Theorem iundif2 4779
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4766 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 3779 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 3229 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 3280 . . . 4 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
4 rexnal 3182 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∀𝑥𝐴 𝑦𝐶)
5 vex 3394 . . . . . . 7 𝑦 ∈ V
6 eliin 4717 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
75, 6ax-mp 5 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
84, 7xchbinxr 326 . . . . 5 (∃𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
98anbi2i 611 . . . 4 ((𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
102, 3, 93bitri 288 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
11 eliun 4716 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
12 eldif 3779 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1310, 11, 123bitr4i 294 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1413eqriv 2803 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  Vcvv 3391  cdif 3766   ciun 4712   ciin 4713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-iun 4714  df-iin 4715
This theorem is referenced by:  iuncld  21059  pnrmopn  21357  alexsublem  22057  bcth3  23336  iundifdifd  29701  iundifdif  29702
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