Description: A more general version of cbvrabv . Version of cbvrabv2 with a disjoint variable condition, which does not require ax-13 . (Contributed by Glauco Siliprandi, 23-Oct-2021) (Revised by GG, 14-Aug-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvrabv2w.1 | ⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 ) | |
| cbvrabv2w.2 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | ||
| Assertion | cbvrabv2w | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∈ 𝐵 ∣ 𝜓 } |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvrabv2w.1 | ⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 ) | |
| 2 | cbvrabv2w.2 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 3 | id | ⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) | |
| 4 | 3 1 | eleq12d | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) ) |
| 5 | 4 2 | anbi12d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) ) |
| 6 | 5 | cbvabv | ⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑦 ∣ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) } |
| 7 | df-rab | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } | |
| 8 | df-rab | ⊢ { 𝑦 ∈ 𝐵 ∣ 𝜓 } = { 𝑦 ∣ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) } | |
| 9 | 6 7 8 | 3eqtr4i | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∈ 𝐵 ∣ 𝜓 } |