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Theorem rspn0 4227
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 4224 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3184 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1890 . . . 4 𝑥𝜑
42, 3nfim 1876 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3170 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2180 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 218 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1759  wcel 2079  wne 2982  wral 3103  c0 4206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-dif 3857  df-nul 4207
This theorem is referenced by:  hashge2el2dif  13672  rmodislmodlem  19379  rmodislmod  19380  scmatf1  20812  fusgrregdegfi  27022  rusgr1vtxlem  27040  upgrewlkle2  27059  ralralimp  42947
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