Axiom ax-16 2144

Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1616 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 in set.mm), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1616; see Theorem ax16 2045. Alternately, ax-17 1616 becomes logically redundant in the presence of this axiom, but without ax-17 1616 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-16 2144 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1616, 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This axiom is obsolete and should no longer be used.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 2045. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-16	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Detailed syntax breakdown of Axiom ax-16

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar x
2		vy	. . . 4 setvar y
3	1, 2	weq 1643	. . 3 wff x = y
4	3, 1	wal 1540	. 2 wff ∀x x = y
5		wph	. . 3 wff φ
6	5, 1	wal 1540	. . 3 wff ∀xφ
7	5, 6	wi 4	. 2 wff (φ → ∀xφ)
8	4, 7	wi 4	1 wff (∀x x = y → (φ → ∀xφ))

This axiom is referenced by:  ax17eq  2183  a16g-o  2186  ax17el  2189  ax10-16  2190