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Theorem moim 2668
Description: "At most one" 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 1891 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
32eximdv 1998 . 2 (∀𝑥(𝜑𝜓) → (∃𝑦𝑥(𝜓𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 mo2v 2625 . 2 (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
5 mo2v 2625 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
63, 4, 53imtr4g 285 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852  ∃*wmo 2619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-eu 2622  df-mo 2623
This theorem is referenced by:  moimi  2669  euimmo  2671  moexex  2690  rmoim  3559  rmoimi2  3561  disjss1  4760  disjss3  4785  funmo  6047  brdom6disj  9556  uptx  21649  taylf  24335  moimd  29666  ssrmo  29673  funressnfv  41728
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