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 ) ) ) |
| 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 ) ) ) |