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Theorem fipjust 38390
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 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 3950 . . 3 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
21eleq1d 2824 . 2 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝐴 ↔ (𝑥𝑣) ∈ 𝐴))
3 ineq2 3951 . . 3 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
43eleq1d 2824 . 2 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝐴 ↔ (𝑥𝑦) ∈ 𝐴))
52, 4cbvral2v 3318 1 (∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2139  wral 3050  cin 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-in 3722
This theorem is referenced by: (None)
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