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Theorem rspn0 3892
 Description: Specialization for restricted generalization with a nonempty set. (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 3890 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 2925 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1830 . . . 4 𝑥𝜑
42, 3nfim 1813 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 2913 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2073 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 206 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  hashge2el2dif  13117  scmatf1  20156  usgfiregdegfi  26438  ralralimp  40309  fusgrregdegfi  40769  rusgr1vtxlem  40787  upgrewlkle2  40808
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