Description: Conversion of implicit substitution to explicit substitution (deduction version of sbievw ). Version of sbied and sbiedv with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Gino Giotto, 29-Jan-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sbiedvw.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| Assertion | sbiedvw | ⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbiedvw.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | sbrimvw | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) | |
| 3 | 1 | expcom | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ) |
| 4 | 3 | pm5.74d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) ) |
| 5 | 4 | sbievw | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) |
| 6 | 2 5 | bitr3i | ⊢ ( ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) |
| 7 | 6 | pm5.74ri | ⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |