Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind is derived from e2ebindVD . (Contributed by Alan Sare, 27-Nov-2014) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | e2ebind | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biidd | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜑 ) ) | |
| 2 | 1 | drex1 | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 𝜑 ↔ ∃ 𝑥 𝜑 ) ) |
| 3 | 2 | drex2 | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 ) ) |
| 4 | excom | ⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜑 ) | |
| 5 | 3 4 | bitrdi | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ∃ 𝑦 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜑 ) ) |
| 6 | nfe1 | ⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑 | |
| 7 | 6 | 19.9 | ⊢ ( ∃ 𝑦 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 𝜑 ) |
| 8 | 5 7 | bitr3di | ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 𝜑 ) ) |
| 9 | 8 | aecoms | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 𝜑 ) ) |