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Theorem eliin2 38823
Description: Membership in indexed intersection. See eliincex 38817 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4498 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 38809). (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 2761 . 2 𝑥𝐵
21eliin2f 38809 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 1987  wne 2790  wral 2908  c0 3897   ciin 4493
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-nul 3898  df-iin 4495
This theorem is referenced by:  eliuniin2  38828  allbutfi  39115
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