Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reuimrmo Structured version   Visualization version   GIF version

Theorem reuimrmo 39827
 Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2510. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
reuimrmo (∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Proof of Theorem reuimrmo
StepHypRef Expression
1 reurmo 3138 . 2 (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜓)
2 rmoim 3374 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
31, 2syl5 33 1 (∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wral 2896  ∃!wreu 2898  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904 This theorem is referenced by:  2reurmo  39831
 Copyright terms: Public domain W3C validator