Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspn0 Structured version   Visualization version   GIF version

Theorem rspn0 4081
 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 4078 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3090 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1995 . . . 4 𝑥𝜑
42, 3nfim 1977 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3078 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2242 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 207 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∅c0 4063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-nul 4064 This theorem is referenced by:  hashge2el2dif  13464  rmodislmodlem  19140  rmodislmod  19141  scmatf1  20555  fusgrregdegfi  26700  rusgr1vtxlem  26718  upgrewlkle2  26737  ralralimp  41819
 Copyright terms: Public domain W3C validator