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

Theorem rmoimi 3684
 Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
rmoimi.1 (𝜑𝜓)
Assertion
Ref Expression
rmoimi (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)

Proof of Theorem rmoimi
StepHypRef Expression
1 rmoimi.1 . . 3 (𝜑𝜓)
21a1i 11 . 2 (𝑥𝐴 → (𝜑𝜓))
32rmoimia 3683 1 (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  ∃*wrmo 3112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-ral 3114  df-rmo 3117 This theorem is referenced by:  2rexreu  3704  2sqreunnlem1  26037  disjin  30353  disjin2  30354  addinvcom  39565
 Copyright terms: Public domain W3C validator