Theorem exmonim 2070

Description: There is at most one of something which does not exist. Unlike exmodc 2069 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)

Assertion

Ref	Expression
exmonim	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem exmonim

Step	Hyp	Ref	Expression
1		pm2.21 612	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 2023	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 133	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-mo 2023

This theorem is referenced by: (None)