Theorem n0limd 32562

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 4284	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3	1, 2	sylib 219	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
4		n0limd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
5	3, 4	exlimddv 1938	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1782   ∈ wcel 2115   ≠ wne 2931  ∅c0 4264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-9 2125  ax-ext 2708

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1546  df-fal 1556  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-ne 2932  df-dif 3889  df-nul 4265

This theorem is referenced by:  fconst7v  32715  ricnzr1  33372  ricdomn1  33373  dimlssid  33819  fldextrspunlem1  33862