Theorem rspn0 4309

Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2175, ax-12 2212. (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 4305	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		df-ral 3077	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		exim 1854	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑))
4		ax5e 1932	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
5	3, 4	syl6com 37	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝜑))
6	2, 5	biimtrid 244	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	1, 6	sylbi 219	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1558  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3077  df-dif 3907  df-nul 4286

This theorem is referenced by:  r19.3rzv  4457  hashge2el2dif  14493  rmodislmodlem  20996  rmodislmod  20997  scmatf1  22591  fusgrregdegfi  29770  rusgr1vtxlem  29788  upgrewlkle2  29807  zarclsiin  34168  ralralimp  47872