Axiom ax-16 1247

Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1007 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 2831), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1007; see theorem ax16 1246. Alternately, ax-17 1007 becomes logically redundant in the presence of this axiom, but without ax-17 1007 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 1247 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1007, which might be easier to study for some theoretical purposes.

Assertion

Ref	Expression
ax-16	|- (A.x x = y -> (ph -> A.xph))

Distinct variable group:   x,y

Detailed syntax breakdown of Axiom ax-16

Step	Hyp	Ref	Expression
1		vx	. . . . 5 set x
2	1	cv 991	. . . 4 class x
3		vy	. . . . 5 set y
4	3	cv 991	. . . 4 class y
5	2, 4	wceq 992	. . 3 wff x = y
6	5, 1	wal 990	. 2 wff A.x x = y
7		wph	. . 3 wff ph
8	7, 1	wal 990	. . 3 wff A.xph
9	7, 8	wi 3	. 2 wff (ph -> A.xph)
10	6, 9	wi 3	1 wff (A.x x = y -> (ph -> A.xph))

This axiom is referenced by:  ax17eq 1248  ax11v 1303  a16g 1314  hbs1 1371  hbsb 1372  sbal1 1385  ax17el 1400  exists2 1499  hbab 1509  hbabd 1510

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