Description: Change bound variable of a proper substitution into a class. General version of cbvcsbdavw . Deduction form. (Contributed by GG, 14-Aug-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvcsbdavw2.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| cbvcsbdavw2.2 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 ) | ||
| Assertion | cbvcsbdavw2 | ⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑦 ⦌ 𝐷 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvcsbdavw2.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| 2 | cbvcsbdavw2.2 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 ) | |
| 3 | 2 | eleq2d | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) ) |
| 4 | 1 3 | cbvsbcdavw2 | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐶 ↔ [ 𝐵 / 𝑦 ] 𝑡 ∈ 𝐷 ) ) |
| 5 | 4 | abbidv | ⊢ ( 𝜑 → { 𝑡 ∣ [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐶 } = { 𝑡 ∣ [ 𝐵 / 𝑦 ] 𝑡 ∈ 𝐷 } ) |
| 6 | df-csb | ⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = { 𝑡 ∣ [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐶 } | |
| 7 | df-csb | ⊢ ⦋ 𝐵 / 𝑦 ⦌ 𝐷 = { 𝑡 ∣ [ 𝐵 / 𝑦 ] 𝑡 ∈ 𝐷 } | |
| 8 | 5 6 7 | 3eqtr4g | ⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑦 ⦌ 𝐷 ) |