Description: Change bound variable in a restricted description binder. Version of cbvriotav with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013) (Revised by Gino Giotto, 30-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvriotavw.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | cbvriotavw | ⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑦 ∈ 𝐴 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvriotavw.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 | cbviotavw | ⊢ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) |
| 5 | df-riota | ⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
| 6 | df-riota | ⊢ ( ℩ 𝑦 ∈ 𝐴 𝜓 ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) | |
| 7 | 4 5 6 | 3eqtr4i | ⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑦 ∈ 𝐴 𝜓 ) |