Metamath Proof Explorer


Theorem sbalex

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 = 𝑡𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) )

Proof

Step Hyp Ref Expression
1 nfa1 𝑥𝑥 ( 𝑥 = 𝑡𝜑 )
2 ax12v2 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )
3 2 imp ( ( 𝑥 = 𝑡𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) )
4 1 3 exlimi ( ∃ 𝑥 ( 𝑥 = 𝑡𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) )
5 equs4v ( ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑡𝜑 ) )
6 4 5 impbii ( ∃ 𝑥 ( 𝑥 = 𝑡𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) )