Description: Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv . (Contributed by GG, 1-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvcsbvw2.1 | ⊢ 𝐴 = 𝐵 | |
| cbvcsbvw2.2 | ⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 ) | ||
| Assertion | cbvcsbvw2 | ⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑦 ⦌ 𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvcsbvw2.1 | ⊢ 𝐴 = 𝐵 | |
| 2 | cbvcsbvw2.2 | ⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 ) | |
| 3 | 2 | eleq2d | ⊢ ( 𝑥 = 𝑦 → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) ) |
| 4 | 1 3 | cbvsbcvw2 | ⊢ ( [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐶 ↔ [ 𝐵 / 𝑦 ] 𝑡 ∈ 𝐷 ) |
| 5 | 4 | abbii | ⊢ { 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 | 3eqtr4i | ⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑦 ⦌ 𝐷 |