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Axiom ax-16 1606
Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1413 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, but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1413; see theorem ax16 1605. Alternately, ax-17 1413 becomes logically redundant in the presence of this axiom, but without ax-17 1413 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 1606 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1413, which might be easier to study for some theoretical purposes.

Assertion
Ref Expression
ax-16
Distinct variable group:   ,
Allowed substitution hints:   (,)

Detailed syntax breakdown of Axiom ax-16
StepHypRef Expression
1 vx . . . 4
2 vy . . . 4
31, 2weq 1399 . . 3
43, 1wal 1322 . 2
5 wph . . 3
65, 1wal 1322 . . 3
75, 6wi 4 . 2
84, 7wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  ax17eq  1607  ax11v  1665  a16g  1675  hbs1  1737  hbsb  1740  hbsbd  1741  ax17el  1770  exists2  1876  hbabd  1887
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