Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004) (Revised by Gino Giotto, 30-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvralvw.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | cbvreuvw | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ∈ 𝐴 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvralvw.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | eleq1w | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) | |
| 3 | 2 1 | anbi12d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) ) |
| 4 | 3 | cbveuvw | ⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) |
| 5 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
| 6 | df-reu | ⊢ ( ∃! 𝑦 ∈ 𝐴 𝜓 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) | |
| 7 | 4 5 6 | 3bitr4i | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ∈ 𝐴 𝜓 ) |