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Theorem fipjust 40717
Description: A definition of the finite intersection property of a class based on closure under pairwise intersection of its elements is independent of the dummy variables. (Contributed by RP, 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 4096 . . 3 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
21eleq1d 2817 . 2 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝐴 ↔ (𝑥𝑣) ∈ 𝐴))
3 ineq2 4097 . . 3 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
43eleq1d 2817 . 2 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝐴 ↔ (𝑥𝑦) ∈ 𝐴))
52, 4cbvral2vw 3362 1 (∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  wral 3053  cin 3842
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-in 3850
This theorem is referenced by: (None)
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