Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb5 ) or a nonfreeness hypothesis ( sb5f ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb1v when possible. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sb1 | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spsbe | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 𝜑 ) | |
| 2 | pm3.2 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) | |
| 3 | 2 | aleximi | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
| 4 | 1 3 | syl5 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
| 5 | sb3b | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) | |
| 6 | 5 | biimpd | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
| 7 | 4 6 | pm2.61i | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |