Theorem rspn0 4227

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 4224	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		nfra1 3184	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
3		nfv 1890	. . . 4 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1876	. . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑)
5		rsp 3170	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
6	5	com12 32	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	4, 6	exlimi 2180	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
8	1, 7	sylbi 218	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∃wex 1759   ∈ wcel 2079   ≠ wne 2982  ∀wral 3103  ∅c0 4206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-dif 3857  df-nul 4207

This theorem is referenced by:  hashge2el2dif  13672  rmodislmodlem  19379  rmodislmod  19380  scmatf1  20812  fusgrregdegfi  27022  rusgr1vtxlem  27040  upgrewlkle2  27059  ralralimp  42947