Theorem nexmo1 36029

Description: If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.)

Assertion

Ref	Expression
nexmo1	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmo1

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		moeu 2584	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 237	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1786  ∃*wmo 2538  ∃!weu 2569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-mo 2540  df-eu 2570

This theorem is referenced by: (None)