MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspn0OLD Structured version   Visualization version   GIF version

Theorem rspn0OLD 4240
Description: Obsolete version of rspn0 4239 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 4233 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 nfra1 3130 . . . 4 𝑥𝑥𝐴 𝜑
3 nfv 1920 . . . 4 𝑥𝜑
42, 3nfim 1902 . . 3 𝑥(∀𝑥𝐴 𝜑𝜑)
5 rsp 3117 . . . 4 (∀𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
65com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
74, 6exlimi 2218 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
81, 7sylbi 220 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1786  wcel 2113  wne 2934  wral 3053  c0 4209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-10 2144  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-ne 2935  df-ral 3058  df-dif 3844  df-nul 4210
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator