Axiom ax-11 1309

Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A.x(x = y -> ph) is a way of expressing "y substituted for x in wff ph " (cf. sb6 1644). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-11o 1588 ("o" for "old") and was replaced with this shorter ax-11 1309 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1587. Conversely, this axiom is proved from ax-11o 1588 as theorem ax11 1589.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1588) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

Interestingly, if the wff expression substituted for ph contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1588 (from which the ax-11 1309 instance follows by theorem ax11 1589.) The proof is by induction on formula length, using ax11eq 1754 and ax11el 1755 for the basis steps and ax11indn 1757, ax11indi 1758, and ax11inda 1762 for the induction steps.

See also ax11v 1642 and ax11v2 1585 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

Assertion

Ref	Expression
ax-11	|- (x = y -> (A.yph -> A.x(x = y -> ph)))

Detailed syntax breakdown of Axiom ax-11

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 1297	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 1297	. . 3 class y
5	2, 4	wceq 1298	. 2 wff x = y
6		wph	. . . 4 wff ph
7	6, 3	wal 1296	. . 3 wff A.yph
8	5, 6	wi 3	. . . 4 wff (x = y -> ph)
9	8, 1	wal 1296	. . 3 wff A.x(x = y -> ph)
10	7, 9	wi 3	. 2 wff (A.yph -> A.x(x = y -> ph))
11	5, 10	wi 3	1 wff (x = y -> (A.yph -> A.x(x = y -> ph)))

This axiom is referenced by:  ax4 1318  ax10o 1499  equs5a 1566  equs5e 1567  ax11o 1587

Copyright terms: Public domain