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Theorem rspn0 4266
 Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2142, ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.)
Assertion
Ref Expression
rspn0 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem rspn0
StepHypRef Expression
1 n0 4260 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 df-ral 3111 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 exim 1835 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥𝜑))
4 ax5e 1913 . . . 4 (∃𝑥𝜑𝜑)
53, 4syl6com 37 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥(𝑥𝐴𝜑) → 𝜑))
62, 5syl5bi 245 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
71, 6sylbi 220 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∅c0 4243 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-dif 3884  df-nul 4244 This theorem is referenced by:  hashge2el2dif  13834  rmodislmodlem  19694  rmodislmod  19695  scmatf1  21136  fusgrregdegfi  27359  rusgr1vtxlem  27377  upgrewlkle2  27396  zarclsiin  31224  ralralimp  43829
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