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Theorem fipjust 40059
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 4159 . . 3 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
21eleq1d 2895 . 2 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝐴 ↔ (𝑥𝑣) ∈ 𝐴))
3 ineq2 4161 . . 3 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
43eleq1d 2895 . 2 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝐴 ↔ (𝑥𝑦) ∈ 𝐴))
52, 4cbvral2vw 3440 1 (∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2114  wral 3125  cin 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ral 3130  df-rab 3134  df-in 3920
This theorem is referenced by: (None)
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