Description: Conversion of implicit substitution to explicit class substitution. This version of csbie avoids a disjointness condition on x , A and x , D by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | csbie2g.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
| csbie2g.2 | ⊢ ( 𝑦 = 𝐴 → 𝐶 = 𝐷 ) | ||
| Assertion | csbie2g | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐷 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbie2g.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
| 2 | csbie2g.2 | ⊢ ( 𝑦 = 𝐴 → 𝐶 = 𝐷 ) | |
| 3 | df-csb | ⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑧 ∣ [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 } | |
| 4 | 1 | eleq2d | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) ) |
| 5 | 2 | eleq2d | ⊢ ( 𝑦 = 𝐴 → ( 𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷 ) ) |
| 6 | 4 5 | sbcie2g | ⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷 ) ) |
| 7 | 6 | abbi1dv | ⊢ ( 𝐴 ∈ 𝑉 → { 𝑧 ∣ [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 } = 𝐷 ) |
| 8 | 3 7 | eqtrid | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐷 ) |