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Theorem rspn0 4134
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 4131 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3122 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 2010 . . . 4 𝑥𝜑
42, 3nfim 1996 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3110 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2252 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 209 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1875  wcel 2157  wne 2971  wral 3089  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-v 3387  df-dif 3772  df-nul 4116
This theorem is referenced by:  hashge2el2dif  13511  rmodislmodlem  19248  rmodislmod  19249  scmatf1  20663  fusgrregdegfi  26819  rusgr1vtxlem  26837  upgrewlkle2  26856  ralralimp  42133
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