Theorem exmoeub 2606

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 2595	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	baibr 546	1 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  ∃*wmo 2563  ∃!weu 2594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-eu 2595

This theorem is used by:  exmoeu  2607  moeu  2609  euim  2643  fneu  6647