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Theorem nrexrmo 3139
Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
nrexrmo (¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Proof of Theorem nrexrmo
StepHypRef Expression
1 pm2.21 118 . 2 (¬ ∃𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
2 rmo5 3138 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
31, 2sylibr 222 1 (¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wrex 2896  ∃!wreu 2897  ∃*wrmo 2898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195  df-mo 2462  df-rex 2901  df-reu 2902  df-rmo 2903
This theorem is referenced by: (None)
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