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Axiom ax-16 1677
 Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1400 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 1400; see theorem ax16 1676. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1676. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Assertion
Ref Expression
ax-16 (x x = y → (φxφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Detailed syntax breakdown of Axiom ax-16
StepHypRef Expression
1 vx . . . 4 setvar x
2 vy . . . 4 setvar y
31, 2weq 1373 . . 3 wff x = y
43, 1wal 1226 . 2 wff x x = y
5 wph . . 3 wff φ
65, 1wal 1226 . . 3 wff xφ
75, 6wi 4 . 2 wff (φxφ)
84, 7wi 4 1 wff (x x = y → (φxφ))
 Colors of variables: wff set class This axiom is referenced by: (None)
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