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Theorem eliin2f 38758
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 4496 . . 3 (𝐴 ∈ V → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
21adantl 482 . 2 ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 prcnel 3209 . . . 4 𝐴 ∈ V → ¬ 𝐴 𝑥𝐵 𝐶)
43adantl 482 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 𝑥𝐵 𝐶)
5 n0 3912 . . . . . . . . 9 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65biimpi 206 . . . . . . . 8 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
76adantr 481 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦𝐵)
8 prcnel 3209 . . . . . . . . . . 11 𝐴 ∈ V → ¬ 𝐴𝑦 / 𝑥𝐶)
98a1d 25 . . . . . . . . . 10 𝐴 ∈ V → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
109adantl 482 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → ¬ 𝐴𝑦 / 𝑥𝐶))
1110ancld 575 . . . . . . . 8 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦𝐵 → (𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
1211eximdv 1848 . . . . . . 7 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶)))
137, 12mpd 15 . . . . . 6 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦𝐵 ∧ ¬ 𝐴𝑦 / 𝑥𝐶))
14 df-rex 2918 . . . . . 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 2767 . . . . . 6 𝑦𝐵
18 nfv 1845 . . . . . 6 𝑦 ¬ 𝐴𝐶
19 nfcsb1v 3535 . . . . . . . 8 𝑥𝑦 / 𝑥𝐶
2019nfel2 2783 . . . . . . 7 𝑥 𝐴𝑦 / 𝑥𝐶
2120nfn 1782 . . . . . 6 𝑥 ¬ 𝐴𝑦 / 𝑥𝐶
22 csbeq1a 3528 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
2322eleq2d 2689 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝐶𝐴𝑦 / 𝑥𝐶))
2423notbid 308 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝐴𝐶 ↔ ¬ 𝐴𝑦 / 𝑥𝐶))
2516, 17, 18, 21, 24cbvrexf 3159 . . . . 5 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ∃𝑦𝐵 ¬ 𝐴𝑦 / 𝑥𝐶)
2615, 25sylibr 224 . . . 4 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥𝐵 ¬ 𝐴𝐶)
27 rexnal 2994 . . . 4 (∃𝑥𝐵 ¬ 𝐴𝐶 ↔ ¬ ∀𝑥𝐵 𝐴𝐶)
2826, 27sylib 208 . . 3 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥𝐵 𝐴𝐶)
294, 282falsed 366 . 2 ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
302, 29pm2.61dan 831 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wex 1701  wcel 1992  wnfc 2754  wne 2796  wral 2912  wrex 2913  Vcvv 3191  csb 3519  c0 3896   ciin 4491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-nul 3897  df-iin 4493
This theorem is referenced by:  eliin2  38772
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