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|Description: Axiom of Quantified
Equality. One of the equality and substitution axioms
of predicate calculus with equality.
An equivalent way to express this axiom that may be easier to understand is (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax12b 1712). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable. Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-17 1636, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows.
The original version of this axiom was ax-12o 2260 and was replaced with this shorter ax-12 1963 in December 2015. The old axiom is proved from this one as theorem ax12o 2024. Conversely, this axiom is proved from ax-12o 2260 as theorem ax12 2033.
The primary purpose of this axiom is to provide a way to introduce the quantifier ∀𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds.
Although this version is shorter, the original version ax12o 2024 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of ax12o 2024 is in dvelimh 2087 which converts a distinct variable pair to the distinctor antecendent ¬ ∀𝑥𝑥 = 𝑦.
This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to theorem ax12v 1964 and use only ax12v 1964 for further derivations. Thus, ax12v 1964 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1964 or ax12o 2024.
This axiom scheme is logically redundant (see ax12w 1748) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)
|ax-12||⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))|
|1||vx||. . . 4 set 𝑥|
|2||vy||. . . 4 set 𝑦|
|3||1, 2||weq 1663||. . 3 wff 𝑥 = 𝑦|
|4||3||wn 3||. 2 wff ¬ 𝑥 = 𝑦|
|5||vz||. . . 4 set 𝑧|
|6||2, 5||weq 1663||. . 3 wff 𝑦 = 𝑧|
|7||6, 1||wal 1556||. . 3 wff ∀𝑥 𝑦 = 𝑧|
|8||6, 7||wi 4||. 2 wff (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)|
|9||4, 8||wi 4||1 wff (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))|
|Colors of variables: wff set class|
|This axiom is referenced by: ax12v 1964|
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