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Theorem moim 2605
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 1817 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
32eximdv 1918 . 2 (∀𝑥(𝜑𝜓) → (∃𝑦𝑥(𝜓𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 df-mo 2601 . 2 (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
5 df-mo 2601 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
63, 4, 53imtr4g 299 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  ∃*wmo 2599
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-ex 1782  df-mo 2601
This theorem is referenced by:  moimdv  2607  mobi  2608  euimmo  2680  moexexlem  2691  rmoim  3682  rmoimi2  3685  ssrmof  3983  disjss3  5032  funmo  6344  uptx  22233  taylf  24959
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