Description: Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in Megill p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993) Revise df-sb . (Revised by BJ, 30-Dec-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbequ | ⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equequ2 | ⊢ ( 𝑥 = 𝑦 → ( 𝑢 = 𝑥 ↔ 𝑢 = 𝑦 ) ) | |
| 2 | 1 | imbi1d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑢 = 𝑥 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ↔ ( 𝑢 = 𝑦 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ) ) |
| 3 | 2 | albidv | ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑢 ( 𝑢 = 𝑥 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ↔ ∀ 𝑢 ( 𝑢 = 𝑦 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ) ) |
| 4 | df-sb | ⊢ ( [ 𝑥 / 𝑧 ] 𝜑 ↔ ∀ 𝑢 ( 𝑢 = 𝑥 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ) | |
| 5 | df-sb | ⊢ ( [ 𝑦 / 𝑧 ] 𝜑 ↔ ∀ 𝑢 ( 𝑢 = 𝑦 → ∀ 𝑧 ( 𝑧 = 𝑢 → 𝜑 ) ) ) | |
| 6 | 3 4 5 | 3bitr4g | ⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] 𝜑 ) ) |