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Theorem rspn0 3912
 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 3909 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 2936 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1840 . . . 4 𝑥𝜑
42, 3nfim 1822 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 2924 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2084 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 207 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∅c0 3893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3559  df-nul 3894 This theorem is referenced by:  hashge2el2dif  13203  scmatf1  20259  fusgrregdegfi  26342  rusgr1vtxlem  26360  upgrewlkle2  26379  ralralimp  40608
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