Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 27-Nov-2005) (Revised by Mario Carneiro, 11-Nov-2016) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sbcco3g.1 | ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) | |
| Assertion | csbco3g | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐷 = ⦋ 𝐶 / 𝑦 ⦌ 𝐷 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcco3g.1 | ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) | |
| 2 | csbnestg | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐷 = ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ⦌ 𝐷 ) | |
| 3 | elex | ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) | |
| 4 | nfcvd | ⊢ ( 𝐴 ∈ V → Ⅎ 𝑥 𝐶 ) | |
| 5 | 4 1 | csbiegf | ⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) |
| 6 | 3 5 | syl | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) |
| 7 | 6 | csbeq1d | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ⦌ 𝐷 = ⦋ 𝐶 / 𝑦 ⦌ 𝐷 ) |
| 8 | 2 7 | eqtrd | ⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐷 = ⦋ 𝐶 / 𝑦 ⦌ 𝐷 ) |