Theorem rspn0 4311

Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2179, ax-12 2216. (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 3082	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		exim 1867	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑))
4		ax5e 1945	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
5	3, 4	syl6com 38	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝜑))
6	2, 5	biimtrid 245	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
7	1, 6	sylbi 220	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-ne 2961  df-ral 3082  df-dif 3909  df-nul 4287

This theorem is used by:  r19.3rzv  4466  hashge2el2dif  14537  rmodislmodlem  21102  rmodislmod  21103  scmatf1  22740  fusgrregdegfi  29979  rusgr1vtxlem  29997  upgrewlkle2  30016  zarclsiin  34327  ralralimp  48075