Theorem exmoeu 2502

Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)

Assertion

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

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 2475	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2	1	biimpi 206	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3	2	com12 32	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4		exmo 2495	. . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
5	4	ori 390	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
6	5	con1i 144	. . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
7		euex 2494	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
8	6, 7	ja 173	. 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
9	3, 8	impbii 199	1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1704  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)