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Theorem eliin2 42187
Description: Membership in indexed intersection. See eliincex 42182 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4883 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 42176). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
eliin2 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliin2
StepHypRef Expression
1 nfcv 2899 . 2 𝑥𝐵
21eliin2f 42176 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2113  wne 2934  wral 3053  c0 4209   ciin 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-nul 4210  df-iin 4881
This theorem is referenced by:  eliuniin2  42191  allbutfi  42455
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