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Metamath Proof Explorer

Theorem eu2

Description: An alternate way of defining existential uniqueness. Definition 6.10 of TakeutiZaring p. 26. (Contributed by NM, 8-Jul-1994) (Proof shortened by Wolf Lammen, 2-Dec-2018)

Ref Expression
Hypothesis eu2.nf ⊢ Ⅎ y φ
Assertion eu2 ⊢ ∃! x φ ↔ ∃ x φ ∧ ∀ x ∀ y φ ∧ y x φ → x = y

Proof

Step Hyp Ref Expression
1 eu2.nf ⊢ Ⅎ y φ
2 df-eu ⊢ ∃! x φ ↔ ∃ x φ ∧ ∃* x φ
3 1 mo3 ⊢ ∃* x φ ↔ ∀ x ∀ y φ ∧ y x φ → x = y
4 3 anbi2i ⊢ ∃ x φ ∧ ∃* x φ ↔ ∃ x φ ∧ ∀ x ∀ y φ ∧ y x φ → x = y
5 2 4 bitri ⊢ ∃! x φ ↔ ∃ x φ ∧ ∀ x ∀ y φ ∧ y x φ → x = y