Theorem rspn0 4134

Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.)

Assertion

Ref	Expression
rspn0	⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem rspn0

Step	Hyp	Ref	Expression
1		n0 4131	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		nfra1 3122	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
3		nfv 2010	. . . 4 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1996	. . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑)
5		rsp 3110	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
6	5	com12 32	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	4, 6	exlimi 2252	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
8	1, 7	sylbi 209	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∃wex 1875   ∈ wcel 2157   ≠ wne 2971  ∀wral 3089  ∅c0 4115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-v 3387  df-dif 3772  df-nul 4116

This theorem is referenced by:  hashge2el2dif  13511  rmodislmodlem  19248  rmodislmod  19249  scmatf1  20663  fusgrregdegfi  26819  rusgr1vtxlem  26837  upgrewlkle2  26856  ralralimp  42133