Theorem n0limd 32441

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 4301	. . 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 1936	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780   ∈ wcel 2110   ≠ wne 2926  ∅c0 4281

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 1968  ax-7 2009  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-ne 2927  df-dif 3903  df-nul 4282

This theorem is referenced by:  fconst7v  32593  dimlssid  33635  fldextrspunlem1  33678