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Theorem rspn0 4362
Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2139, ax-12 2175. (Revised by GG, 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 4359 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 df-ral 3060 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 exim 1831 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥𝜑))
4 ax5e 1910 . . . 4 (∃𝑥𝜑𝜑)
53, 4syl6com 37 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥(𝑥𝐴𝜑) → 𝜑))
62, 5biimtrid 242 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
71, 6sylbi 217 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1776  wcel 2106  wne 2938  wral 3059  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ne 2939  df-ral 3060  df-dif 3966  df-nul 4340
This theorem is referenced by:  hashge2el2dif  14516  rmodislmodlem  20944  rmodislmod  20945  rmodislmodOLD  20946  scmatf1  22553  fusgrregdegfi  29602  rusgr1vtxlem  29620  upgrewlkle2  29639  zarclsiin  33832  ralralimp  47228
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