Theorem exmoeu 2611

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 2610	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
2	1	biimpd 232	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
3		nexmo 2571	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
4	3	con1i 148	. . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
5		euex 2607	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
6	4, 5	ja 188	. 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
7	2, 6	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  ∃*wmo 2567  ∃!weu 2598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599

This theorem is used by:  tfsconcatlem  44123