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Theorem eliin2f 42327
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 4909 . . 3 (𝐴 ∈ V → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
21adantl 485 . 2 ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 prcnel 3431 . . . 4 𝐴 ∈ V → ¬ 𝐴 𝑥𝐵 𝐶)
43adantl 485 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 𝑥𝐵 𝐶)
5 n0 4261 . . . . . . . . 9 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65biimpi 219 . . . . . . . 8 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
76adantr 484 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦𝐵)
8 prcnel 3431 . . . . . . . . . . 11 𝐴 ∈ V → ¬ 𝐴𝑦 / 𝑥𝐶)
98a1d 25 . . . . . . . . . 10 𝐴 ∈ V → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
109adantl 485 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
1110ancld 554 . . . . . . . 8 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → (𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
1211eximdv 1925 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
137, 12mpd 15 . . . . . 6 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
14 df-rex 3067 . . . . . 6 (∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶 ↔ ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
1513, 14sylibr 237 . . . . 5 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
16 eliin2f.1 . . . . . 6 𝑥𝐵
17 nfcv 2904 . . . . . 6 𝑦𝐵
18 nfv 1922 . . . . . 6 𝑦 ¬ 𝐴𝐶
19 nfcsb1v 3836 . . . . . . . 8 𝑥𝑦 / 𝑥𝐶
2019nfel2 2922 . . . . . . 7 𝑥 𝐴𝑦 / 𝑥𝐶
2120nfn 1865 . . . . . 6 𝑥 ¬ 𝐴𝑦 / 𝑥𝐶
22 csbeq1a 3825 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
2322eleq2d 2823 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝐶𝐴𝑦 / 𝑥𝐶))
2423notbid 321 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝐴𝐶 ↔ ¬ 𝐴𝑦 / 𝑥𝐶))
2516, 17, 18, 21, 24cbvrexfw 3346 . . . . 5 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
2615, 25sylibr 237 . . . 4 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥𝐵 ¬ 𝐴𝐶)
27 rexnal 3160 . . . 4 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ¬ ∀𝑥𝐵 𝐴𝐶)
2826, 27sylib 221 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥𝐵 𝐴𝐶)
294, 282falsed 380 . 2 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
302, 29pm2.61dan 813 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wex 1787  wcel 2110  wnfc 2884  wne 2940  wral 3061  wrex 3062  Vcvv 3408  csb 3811  c0 4237   ciin 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-nul 4238  df-iin 4907
This theorem is referenced by:  eliin2  42338
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