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

Theorem rmoim 3374
 Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
rmoim (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Proof of Theorem rmoim
StepHypRef Expression
1 df-ral 2901 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 imdistan 721 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32albii 1737 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
41, 3bitri 263 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 moim 2507 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥(𝑥𝐴𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
6 df-rmo 2904 . . 3 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
7 df-rmo 2904 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
85, 6, 73imtr4g 284 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
94, 8sylbi 206 1 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896  ∃*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-rmo 2904 This theorem is referenced by:  rmoimia  3375  2rmorex  3379  disjss2  4556  catideu  16159  evlseu  19337  frlmup4  19959  2ndcdisj  21069  poimirlem18  32597  poimirlem21  32600  reuimrmo  39827  2reurex  39830
 Copyright terms: Public domain W3C validator