Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb6 ) or a nonfreeness hypothesis ( sb6f ). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 26-Jul-2023) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sb2 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.27 | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) ) | |
| 2 | 1 | al2imi | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 𝜑 ) ) |
| 3 | stdpc4 | ⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) | |
| 4 | 2 3 | syl6 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) ) |
| 5 | sb4b | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) | |
| 6 | 5 | biimprd | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) ) |
| 7 | 4 6 | pm2.61i | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) |