Theorem n0limd 4300

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 4299	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
4		n0limd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
5	3, 4	exlimddv 1967	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-ne 2956  df-dif 4047  df-nul 4279

This theorem is used by:  tglnpt4  29123  angmgmaddeu1  29379  perpprlng  29428  prlngmolem2  29431  prlngplngtr  29437  quadcgrprlng  29444  fconst7v  33256  ricnzr1  33891  ricdomn1  33892  dflringlem3  34070  dflring4  34072  dimlssid  34306  fldextrspunlem1  34349