Description: Obsolete version of sbequi as of 7-Jul-2023. Version of sbequi with disjoint variable conditions, not requiring ax-13 . (Contributed by Wolf Lammen, 19-Jan-2023) (New usage is discouraged.) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbequivvOLD | ⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equeuclr | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → 𝑧 = 𝑥 ) ) | |
| 2 | 1 | imim1d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑧 = 𝑥 → 𝜑 ) → ( 𝑧 = 𝑦 → 𝜑 ) ) ) |
| 3 | 2 | alimdv | ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑧 ( 𝑧 = 𝑥 → 𝜑 ) → ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) ) ) |
| 4 | sb6 | ⊢ ( [ 𝑥 / 𝑧 ] 𝜑 ↔ ∀ 𝑧 ( 𝑧 = 𝑥 → 𝜑 ) ) | |
| 5 | sb6 | ⊢ ( [ 𝑦 / 𝑧 ] 𝜑 ↔ ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) ) | |
| 6 | 3 4 5 | 3imtr4g | ⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) |