Theorem exmoeu 2581

Description: Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)

Assertion

Ref	Expression
exmoeu	⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		exmoeub 2580	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
2	1	biimpd 228	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
3		nexmo 2541	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
4	3	con1i 147	. . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
5		euex 2577	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
6	4, 5	ja 186	. 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
7	2, 6	impbii 208	1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1782  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)