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Theorem rspn0OLD 4287
Description: Obsolete version of rspn0 4286 as of 28-Jun-2024. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rspn0OLD (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem rspn0OLD
StepHypRef Expression
1 n0 4280 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3144 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1917 . . . 4 𝑥𝜑
42, 3nfim 1899 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3131 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2210 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 216 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1782  wcel 2106  wne 2943  wral 3064  c0 4256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-ne 2944  df-ral 3069  df-dif 3890  df-nul 4257
This theorem is referenced by: (None)
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