Description: Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb .
That both sides of the biconditional express proper substitution is proved by sb5 and sb6 . The implication "to the left" is equs4v and does not require ax-10 nor ax-12 . It also holds without disjoint variable condition if we allow more axioms (see equs4 ). Theorem 6.2 of Quine p. 40. Theorem equs5 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f replaces the disjoint variable condition on x , t with the nonfreeness hypothesis of t in ph . (Contributed by NM, 14-Apr-2008) Revised to use equsexv in place of equsex in order to remove dependency on ax-13 . (Revised by BJ, 20-Dec-2020) Revise to remove dependency on df-sb . (Revised by BJ, 21-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbalex | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfa1 | ⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) | |
| 2 | ax12v2 | ⊢ ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) | |
| 3 | 2 | imp | ⊢ ( ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 4 | 1 3 | exlimi | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 5 | equs4v | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ) | |
| 6 | 4 5 | impbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |