Theorem rspn0 4315

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 4312	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		nfra1 3221	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1897	. . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑)
5		rsp 3207	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
6	5	com12 32	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	4, 6	exlimi 2217	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
8	1, 7	sylbi 219	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∅c0 4293

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-dif 3941  df-nul 4294

This theorem is referenced by:  hashge2el2dif  13841  rmodislmodlem  19703  rmodislmod  19704  scmatf1  21142  fusgrregdegfi  27353  rusgr1vtxlem  27371  upgrewlkle2  27390  ralralimp  43484