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Theorem rspn0 4310
Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
Assertion
Ref Expression
rspn0 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem rspn0
StepHypRef Expression
1 n0 4307 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3216 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1906 . . . 4 𝑥𝜑
42, 3nfim 1888 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3202 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2207 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 218 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1771  wcel 2105  wne 3013  wral 3135  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-dif 3936  df-nul 4289
This theorem is referenced by:  hashge2el2dif  13826  rmodislmodlem  19630  rmodislmod  19631  scmatf1  21068  fusgrregdegfi  27278  rusgr1vtxlem  27296  upgrewlkle2  27315  ralralimp  43354
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