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Theorem eliin2f 39804
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 4677 . . 3 (𝐴 ∈ V → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
21adantl 473 . 2 ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 prcnel 3358 . . . 4 𝐴 ∈ V → ¬ 𝐴 𝑥𝐵 𝐶)
43adantl 473 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 𝑥𝐵 𝐶)
5 n0 4074 . . . . . . . . 9 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65biimpi 206 . . . . . . . 8 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
76adantr 472 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦𝐵)
8 prcnel 3358 . . . . . . . . . . 11 𝐴 ∈ V → ¬ 𝐴𝑦 / 𝑥𝐶)
98a1d 25 . . . . . . . . . 10 𝐴 ∈ V → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
109adantl 473 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
1110ancld 577 . . . . . . . 8 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → (𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
1211eximdv 1995 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
137, 12mpd 15 . . . . . 6 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
14 df-rex 3056 . . . . . 6 (∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶 ↔ ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
1513, 14sylibr 224 . . . . 5 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
16 eliin2f.1 . . . . . 6 𝑥𝐵
17 nfcv 2902 . . . . . 6 𝑦𝐵
18 nfv 1992 . . . . . 6 𝑦 ¬ 𝐴𝐶
19 nfcsb1v 3690 . . . . . . . 8 𝑥𝑦 / 𝑥𝐶
2019nfel2 2919 . . . . . . 7 𝑥 𝐴𝑦 / 𝑥𝐶
2120nfn 1933 . . . . . 6 𝑥 ¬ 𝐴𝑦 / 𝑥𝐶
22 csbeq1a 3683 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
2322eleq2d 2825 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝐶𝐴𝑦 / 𝑥𝐶))
2423notbid 307 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝐴𝐶 ↔ ¬ 𝐴𝑦 / 𝑥𝐶))
2516, 17, 18, 21, 24cbvrexf 3305 . . . . 5 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
2615, 25sylibr 224 . . . 4 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥𝐵 ¬ 𝐴𝐶)
27 rexnal 3133 . . . 4 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ¬ ∀𝑥𝐵 𝐴𝐶)
2826, 27sylib 208 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥𝐵 𝐴𝐶)
294, 282falsed 365 . 2 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
302, 29pm2.61dan 867 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wex 1853  wcel 2139  wnfc 2889  wne 2932  wral 3050  wrex 3051  Vcvv 3340  csb 3674  c0 4058   ciin 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-nul 4059  df-iin 4675
This theorem is referenced by:  eliin2  39816
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