Theorem rspn0 4310

Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2147, ax-12 2185. (Revised by GG, 28-Jun-2024.)

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 4307	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		df-ral 3053	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		exim 1836	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑))
4		ax5e 1914	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
5	3, 4	syl6com 37	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝜑))
6	2, 5	biimtrid 242	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	1, 6	sylbi 217	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-ne 2934  df-ral 3053  df-dif 3906  df-nul 4288

This theorem is referenced by:  r19.3rzv  4458  hashge2el2dif  14415  rmodislmodlem  20892  rmodislmod  20893  scmatf1  22487  fusgrregdegfi  29655  rusgr1vtxlem  29673  upgrewlkle2  29692  zarclsiin  34049  ralralimp  47638