Description: Rederive df-mo from the old definition moeu . (Contributed by Wolf Lammen, 27-May-2019) (Proof modification is discouraged.) Use dfmo instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfmo2 | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | moeu | ⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ) | |
| 2 | eu6 | ⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) | |
| 3 | 2 | imbi2i | ⊢ ( ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ↔ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 4 | dfmoeu | ⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) | |
| 5 | 1 3 4 | 3bitri | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |