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Theorem n0limd 32500
Description: Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.)
Hypotheses
Ref Expression
n0limd.1 (𝜑𝐴 ≠ ∅)
n0limd.2 ((𝜑𝑥𝐴) → 𝜓)
Assertion
Ref Expression
n0limd (𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜓,𝑥

Proof of Theorem n0limd
StepHypRef Expression
1 n0limd.1 . . 3 (𝜑𝐴 ≠ ∅)
2 n0 4358 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
31, 2sylib 218 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
4 n0limd.2 . 2 ((𝜑𝑥𝐴) → 𝜓)
53, 4exlimddv 1932 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1775  wcel 2105  wne 2937  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-ne 2938  df-dif 3965  df-nul 4339
This theorem is referenced by:  dimlssid  33659
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