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Axiom ax-13 1955
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 (see ax13b 1717). 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 to hold, and must have different values and thus cannot be the same object-language variable. Similarly, and cannot be the same object-language variable. Therefore, will not occur in the wff when the first two antecedents hold, so analogous to ax-5 1644, the conclusion follows.

The original version of this axiom was ax-c9 2252 and was replaced with this shorter ax-13 1955 in December 2015. The old axiom is proved from this one as theorem axc9 2016. Conversely, this axiom is proved from ax-c9 2252 as theorem ax13 2025.

The primary purpose of this axiom is to provide a way to introduce the quantifier A. on even when and are substituted with the same variable. In this case, the first antecedent becomes and the axiom still holds.

Although this version is shorter, the original version axc9 2016 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2016 is in dvelimh 2079 which converts a distinct variable pair to the distinctor antecedent . In particular, it is conjectured that it is not possible to prove ax6 1960 from ax6v 1686 without this axiom.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs , and , . To show that, we add these restrictions to theorem ax13v 1956 and use only ax13v 1956 for further derivations. Thus, ax13v 1956 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 1956 or axc9 2016.

This axiom scheme is logically redundant (see ax13w 1746) but is used as an auxiliary axiom to achieve metalogical completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#wal (future)). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Ref Expression

Detailed syntax breakdown of Axiom ax-13
StepHypRef Expression
1 vx . . . 4
2 vy . . . 4
31, 2weq 1671 . . 3
43wn 3 . 2
5 vz . . . 4
62, 5weq 1671 . . 3
76, 1wal 1564 . . 3
86, 7wi 4 . 2
94, 8wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  ax13v  1956
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