Theorem n0limd 32408

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 4319	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
4		n0limd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
5	3, 4	exlimddv 1935	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-ne 2927  df-dif 3920  df-nul 4300

This theorem is referenced by:  dimlssid  33635  fldextrspunlem1  33677