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Axiom ax-16 2096
Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1606 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 4217), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1606; see theorem ax16 1998. Alternately, ax-17 1606 becomes logically redundant in the presence of this axiom, but without ax-17 1606 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 2096 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1606, 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This axiom is obsolete and should no longer be used.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 ax16 1998. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-16  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Detailed syntax breakdown of Axiom ax-16
StepHypRef Expression
1 vx . . . 4  set  x
2 vy . . . 4  set  y
31, 2weq 1633 . . 3  wff  x  =  y
43, 1wal 1530 . 2  wff  A. x  x  =  y
5 wph . . 3  wff  ph
65, 1wal 1530 . . 3  wff  A. x ph
75, 6wi 4 . 2  wff  ( ph  ->  A. x ph )
84, 7wi 4 1  wff  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
This axiom is referenced by:  ax17eq  2135  a16g-o  2138  ax17el  2141  ax10-16  2142
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