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Theorem eliin2f 40739
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 4791 . . 3 (𝐴 ∈ V → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
21adantl 474 . 2 ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 prcnel 3433 . . . 4 𝐴 ∈ V → ¬ 𝐴 𝑥𝐵 𝐶)
43adantl 474 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 𝑥𝐵 𝐶)
5 n0 4191 . . . . . . . . 9 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65biimpi 208 . . . . . . . 8 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
76adantr 473 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦𝐵)
8 prcnel 3433 . . . . . . . . . . 11 𝐴 ∈ V → ¬ 𝐴𝑦 / 𝑥𝐶)
98a1d 25 . . . . . . . . . 10 𝐴 ∈ V → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
109adantl 474 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
1110ancld 543 . . . . . . . 8 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → (𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
1211eximdv 1876 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
137, 12mpd 15 . . . . . 6 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
14 df-rex 3088 . . . . . 6 (∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶 ↔ ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
1513, 14sylibr 226 . . . . 5 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
16 eliin2f.1 . . . . . 6 𝑥𝐵
17 nfcv 2926 . . . . . 6 𝑦𝐵
18 nfv 1873 . . . . . 6 𝑦 ¬ 𝐴𝐶
19 nfcsb1v 3800 . . . . . . . 8 𝑥𝑦 / 𝑥𝐶
2019nfel2 2942 . . . . . . 7 𝑥 𝐴𝑦 / 𝑥𝐶
2120nfn 1819 . . . . . 6 𝑥 ¬ 𝐴𝑦 / 𝑥𝐶
22 csbeq1a 3791 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
2322eleq2d 2845 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝐶𝐴𝑦 / 𝑥𝐶))
2423notbid 310 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝐴𝐶 ↔ ¬ 𝐴𝑦 / 𝑥𝐶))
2516, 17, 18, 21, 24cbvrexf 3372 . . . . 5 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
2615, 25sylibr 226 . . . 4 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥𝐵 ¬ 𝐴𝐶)
27 rexnal 3179 . . . 4 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ¬ ∀𝑥𝐵 𝐴𝐶)
2826, 27sylib 210 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥𝐵 𝐴𝐶)
294, 282falsed 369 . 2 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
302, 29pm2.61dan 800 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wex 1742  wcel 2048  wnfc 2910  wne 2961  wral 3082  wrex 3083  Vcvv 3409  csb 3782  c0 4173   ciin 4787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-nul 4174  df-iin 4789
This theorem is referenced by:  eliin2  40751
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