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Theorem rspn0 4351
Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2137, ax-12 2171. (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 4345 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 df-ral 3062 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 exim 1836 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥𝜑))
4 ax5e 1915 . . . 4 (∃𝑥𝜑𝜑)
53, 4syl6com 37 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥(𝑥𝐴𝜑) → 𝜑))
62, 5biimtrid 241 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
71, 6sylbi 216 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wex 1781  wcel 2106  wne 2940  wral 3061  c0 4321
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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-ne 2941  df-ral 3062  df-dif 3950  df-nul 4322
This theorem is referenced by:  hashge2el2dif  14437  rmodislmodlem  20531  rmodislmod  20532  rmodislmodOLD  20533  scmatf1  22024  fusgrregdegfi  28815  rusgr1vtxlem  28833  upgrewlkle2  28852  zarclsiin  32839  ralralimp  45972
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