Theorem mon 2055

Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)

Assertion

Ref	Expression
mon	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem mon

Step	Hyp	Ref	Expression
1		ax-in2 615	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 2030	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 134	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-mo 2030

This theorem is referenced by:  moexexdc  2110