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Theorem iindif2 4519
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4503 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 4017 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶)))
2 eldif 3549 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32bicomi 212 . . . . 5 ((𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ 𝑦 ∈ (𝐵𝐶))
43ralbii 2962 . . . 4 (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
5 ralnex 2974 . . . . . 6 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ ∃𝑥𝐴 𝑦𝐶)
6 eliun 4454 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
75, 6xchbinxr 323 . . . . 5 (∀𝑥𝐴 ¬ 𝑦𝐶 ↔ ¬ 𝑦 𝑥𝐴 𝐶)
87anbi2i 725 . . . 4 ((𝑦𝐵 ∧ ∀𝑥𝐴 ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
91, 4, 83bitr3g 300 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶)))
10 vex 3175 . . . 4 𝑦 ∈ V
11 eliin 4455 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
1210, 11ax-mp 5 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 3549 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
149, 12, 133bitr4g 301 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶)))
1514eqrdv 2607 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  c0 3873   ciun 4449   ciin 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-nul 3874  df-iun 4451  df-iin 4452
This theorem is referenced by:  iinvdif  4522  iincld  20600  clsval2  20611  mretopd  20653  hauscmplem  20966  cmpfi  20968  sigapildsyslem  29344  saliincl  39004
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