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Axiom ax-13 1999
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 1805). 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 (so they are effectively "distinct variables" even though no $d is present). 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 1704, the conclusion follows. Note that ax-5 1704 cannot prove this directly because it requires $d statements.

The original version of this axiom was ax-c9 2221 and was replaced with this shorter ax-13 1999 in December 2015. The old axiom is proved from this one as theorem axc9 2046. Conversely, this axiom is proved from ax-c9 2221 as theorem ax13 2047.

The primary purpose of this axiom is to provide a way to introduce the quantifier A.x 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 2046 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2046 is in dvelimh 2078 which converts a distinct variable pair to the distinctor antecedent . In particular, it is conjectured that it is not possible to prove ax6 2003 from ax6v 1748 without this axiom.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs x, and , . To show that, we add these restrictions to theorem ax13v 2000 and use only ax13v 2000 for further derivations. Thus, ax13v 2000 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2000 (preferred) or ax13 2047.

This axiom scheme is logically redundant (see ax13w 1832) 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 1733 . . 3
43wn 3 . 2
5 vz . . . 4
62, 5weq 1733 . . 3
76, 1wal 1393 . . 3
86, 7wi 4 . 2
94, 8wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  ax13v  2000
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