Theorem exmoeub 2584

Description: Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004.)

Assertion

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

Proof of Theorem exmoeub

Step	Hyp	Ref	Expression
1		df-eu 2573	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	baibr 541	1 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∃wex 1786  ∃*wmo 2541  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 208  df-an 397  df-eu 2573

This theorem is referenced by:  exmoeu  2585  moeu  2587  euim  2621  fneu  6602