Theorem rspn0 3892

Description: Specialization for restricted generalization with a nonempty set. (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 3890	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		nfra1 2925	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
3		nfv 1830	. . . 4 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1813	. . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑)
5		rsp 2913	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
6	5	com12 32	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	4, 6	exlimi 2073	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
8	1, 7	sylbi 206	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  hashge2el2dif  13117  scmatf1  20156  usgfiregdegfi  26438  ralralimp  40309  fusgrregdegfi  40769  rusgr1vtxlem  40787  upgrewlkle2  40808