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Theorem reuimrmo 3711
Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2701. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
reuimrmo (∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Proof of Theorem reuimrmo
StepHypRef Expression
1 reurmo 3406 . 2 (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜓)
2 rmoim 3706 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
31, 2syl5 34 1 (∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3130  ∃!wreu 3132  ∃*wrmo 3133
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 2622  df-eu 2653  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138
This theorem is referenced by:  2reurmo  3726
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