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

Theorem moim 2542
Description: The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
moim (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem moim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 imim1 83 . . . 4 ((𝜑𝜓) → ((𝜓𝑥 = 𝑦) → (𝜑𝑥 = 𝑦)))
21al2imi 1812 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
32eximdv 1915 . 2 (∀𝑥(𝜑𝜓) → (∃𝑦𝑥(𝜓𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 df-mo 2538 . 2 (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
5 df-mo 2538 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
63, 4, 53imtr4g 296 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1776  ∃*wmo 2536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908
This theorem depends on definitions:  df-bi 207  df-ex 1777  df-mo 2538
This theorem is referenced by:  moimdv  2544  mobi  2545  euimmo  2614  moexexlem  2624  rmoim  3749  rmoimi2  3752  ssrmof  4063  disjss3  5147  funmo  6583  funmoOLD  6584  uptx  23649  taylf  26417
  Copyright terms: Public domain W3C validator