![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > moeuex | Structured version Visualization version GIF version |
Description: Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022.) |
Ref | Expression |
---|---|
moeuex | ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2584 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
2 | 1 | rbaibr 530 | 1 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∃wex 1742 ∃*wmo 2545 ∃!weu 2583 |
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 199 df-an 388 df-eu 2584 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |