Theorem moeu2 38615

Description: Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)

Assertion

Ref	Expression
moeu2	⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))

Proof of Theorem moeu2

Step	Hyp	Ref	Expression
1		moeu 2584	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		imor 854	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
3	1, 2	bitri 275	1 ⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848  ∃wex 1781  ∃*wmo 2538  ∃!weu 2569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-mo 2540  df-eu 2570

This theorem is referenced by:  mopickr  38616