Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 to be a metatheorem and not an axiom). Axiom scheme C16' in Megill p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru ), but nonetheless it is technically necessary as you can see from its uses.
This axiom is redundant if we include ax-5 ; see theorem axc16 . Alternately, ax-5 becomes logically redundant in the presence of this axiom, but without ax-5 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 , which might be easier to study for some theoretical purposes.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc16 . (Contributed by NM, 10-Jan-1993) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-c16 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | ⊢ 𝑥 | |
| 1 | 0 | cv | ⊢ 𝑥 |
| 2 | vy | ⊢ 𝑦 | |
| 3 | 2 | cv | ⊢ 𝑦 |
| 4 | 1 3 | wceq | ⊢ 𝑥 = 𝑦 |
| 5 | 4 0 | wal | ⊢ ∀ 𝑥 𝑥 = 𝑦 |
| 6 | wph | ⊢ 𝜑 | |
| 7 | 6 0 | wal | ⊢ ∀ 𝑥 𝜑 |
| 8 | 6 7 | wi | ⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
| 9 | 5 8 | wi | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) ) |