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Metamath Proof Explorer
Description: Existential uniqueness implies uniqueness through reverse implication.
(Contributed by NM, 22-Apr-1995)
|
|
Ref |
Expression |
|
Assertion |
euimmo |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eumo |
⊢ ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜓 ) |
| 2 |
|
moim |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 𝜓 → ∃* 𝑥 𝜑 ) ) |
| 3 |
1 2
|
syl5 |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) ) |