Description: Composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbcco3gw when possible. (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 | sbcco3g | ⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐶 / 𝑦 ] 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcco3g.1 | ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) | |
| 2 | sbcnestg | ⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) ) | |
| 3 | elex | ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) | |
| 4 | nfcvd | ⊢ ( 𝐴 ∈ V → Ⅎ 𝑥 𝐶 ) | |
| 5 | 4 1 | csbiegf | ⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) |
| 6 | dfsbcq | ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 → ( [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐶 / 𝑦 ] 𝜑 ) ) | |
| 7 | 3 5 6 | 3syl | ⊢ ( 𝐴 ∈ 𝑉 → ( [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐶 / 𝑦 ] 𝜑 ) ) |
| 8 | 2 7 | bitrd | ⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐶 / 𝑦 ] 𝜑 ) ) |