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Theorem eliin2f 41247
Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
eliin2f.1 𝑥𝐵
Assertion
Ref Expression
eliin2f (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliin2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliin 4915 . . 3 (𝐴 ∈ V → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
21adantl 482 . 2 ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 prcnel 3516 . . . 4 𝐴 ∈ V → ¬ 𝐴 𝑥𝐵 𝐶)
43adantl 482 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 𝑥𝐵 𝐶)
5 n0 4307 . . . . . . . . 9 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65biimpi 217 . . . . . . . 8 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
76adantr 481 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦𝐵)
8 prcnel 3516 . . . . . . . . . . 11 𝐴 ∈ V → ¬ 𝐴𝑦 / 𝑥𝐶)
98a1d 25 . . . . . . . . . 10 𝐴 ∈ V → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
109adantl 482 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
1110ancld 551 . . . . . . . 8 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → (𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
1211eximdv 1909 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
137, 12mpd 15 . . . . . 6 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
14 df-rex 3141 . . . . . 6 (∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶 ↔ ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
1513, 14sylibr 235 . . . . 5 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
16 eliin2f.1 . . . . . 6 𝑥𝐵
17 nfcv 2974 . . . . . 6 𝑦𝐵
18 nfv 1906 . . . . . 6 𝑦 ¬ 𝐴𝐶
19 nfcsb1v 3904 . . . . . . . 8 𝑥𝑦 / 𝑥𝐶
2019nfel2 2993 . . . . . . 7 𝑥 𝐴𝑦 / 𝑥𝐶
2120nfn 1848 . . . . . 6 𝑥 ¬ 𝐴𝑦 / 𝑥𝐶
22 csbeq1a 3894 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
2322eleq2d 2895 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝐶𝐴𝑦 / 𝑥𝐶))
2423notbid 319 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝐴𝐶 ↔ ¬ 𝐴𝑦 / 𝑥𝐶))
2516, 17, 18, 21, 24cbvrexfw 3436 . . . . 5 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
2615, 25sylibr 235 . . . 4 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥𝐵 ¬ 𝐴𝐶)
27 rexnal 3235 . . . 4 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ¬ ∀𝑥𝐵 𝐴𝐶)
2826, 27sylib 219 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥𝐵 𝐴𝐶)
294, 282falsed 378 . 2 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
302, 29pm2.61dan 809 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wex 1771  wcel 2105  wnfc 2958  wne 3013  wral 3135  wrex 3136  Vcvv 3492  csb 3880  c0 4288   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289  df-iin 4913
This theorem is referenced by:  eliin2  41259
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