Description: Conversion of implicit substitution to explicit substitution (deduction version of sbiev ). Version of sbied with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-10 . (Revised by Wolf Lammen, 28-Jan-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | sbiedw.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| sbiedw.2 | ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 ) | ||
| sbiedw.3 | ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) ) | ||
| Assertion | sbiedw | ⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbiedw.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| 2 | sbiedw.2 | ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 ) | |
| 3 | sbiedw.3 | ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) ) | |
| 4 | 1 | sbrimv | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) |
| 5 | 1 2 | nfim1 | ⊢ Ⅎ 𝑥 ( 𝜑 → 𝜒 ) |
| 6 | 3 | com12 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ) |
| 7 | 6 | pm5.74d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) ) |
| 8 | 5 7 | sbiev | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) |
| 9 | 4 8 | bitr3i | ⊢ ( ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) |
| 10 | 9 | pm5.74ri | ⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |