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Theorem fipjust 36889
Description: A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
Assertion
Ref Expression
fipjust (∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
Distinct variable group:   𝑣,𝑢,𝑥,𝑦,𝐴

Proof of Theorem fipjust
StepHypRef Expression
1 ineq1 3769 . . 3 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
21eleq1d 2672 . 2 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝐴 ↔ (𝑥𝑣) ∈ 𝐴))
3 ineq2 3770 . . 3 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
43eleq1d 2672 . 2 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝐴 ↔ (𝑥𝑦) ∈ 𝐴))
52, 4cbvral2v 3155 1 (∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wcel 1977  wral 2896  cin 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547
This theorem is referenced by: (None)
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