Axiom ax-16 1206

Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 968 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 2762), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 968; see theorem ax16 1205. Alternately, ax-17 968 becomes logically redundant in the presence of this axiom, but without ax-17 968 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 1206 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 968, which might be easier to study for some theoretical purposes.

Assertion

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

Distinct variable group:   x,y

Detailed syntax breakdown of Axiom ax-16

Step	Hyp	Ref	Expression
1		vx	. . . . 5 set x
2	1	cv 952	. . . 4 class x
3		vy	. . . . 5 set y
4	3	cv 952	. . . 4 class y
5	2, 4	wceq 953	. . 3 wff x = y
6	5, 1	wal 951	. 2 wff ∀x x = y
7		wph	. . 3 wff φ
8	7, 1	wal 951	. . 3 wff ∀xφ
9	7, 8	wi 3	. 2 wff (φ → ∀xφ)
10	6, 9	wi 3	1 wff (∀x x = y → (φ → ∀xφ))

This axiom is referenced by:  ax17eq 1207  ax11v 1260  a16g 1271  hbs1 1327  hbsb 1328  sbal1 1341  ax17el 1354  exists2 1451  hbab 1460  hbabd 1461

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