Theorem n0limd 4307

Description: Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.)

Hypotheses

Ref	Expression
n0limd.1	⊢ (𝜑 → 𝐴 ≠ ∅)
n0limd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Assertion

Ref	Expression
n0limd	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜓,𝑥

Proof of Theorem n0limd

Step	Hyp	Ref	Expression
1		n0limd.1	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		n0 4306	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3	1, 2	sylib 220	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
4		n0limd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
5	3, 4	exlimddv 1956	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1800   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-ne 2959  df-dif 3908  df-nul 4287

This theorem is referenced by:  tglnpt4  28831  fconst7v  32828  ricnzr1  33475  ricdomn1  33476  dflringlem3  33695  dflring4  33697  dimlssid  33931  fldextrspunlem1  33974