Theorem exmonim 1992

Description: There is at most one of something which does not exist. Unlike exmodc 1991 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		pm2.21 579	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 1945	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 132	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-mo 1945

This theorem is referenced by: (None)