Axiom ax-11 1626

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 1943). 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 1893 ("o" for "old") and was replaced with this shorter ax-11 1626 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1892. Conversely, this axiom is proved from ax-11o 1893 as theorem ax11 1894.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1893) 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 1893 (from which the ax-11 1626 instance follows by theorem ax11 1894.) The proof is by induction on formula length, using ax11eq 2047 and ax11el 2048 for the basis steps and ax11indn 2050, ax11indi 2051, and ax11inda 2055 for the induction steps.

See also ax11v 1941 and ax11v2 1890 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 1614	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 1614	. . 3 class y
5	2, 4	wceq 1615	. 2 wff x = y
6		wph	. . . 4 wff ph
7	6, 3	wal 1613	. . 3 wff A.yph
8	5, 6	wi 3	. . . 4 wff (x = y -> ph)
9	8, 1	wal 1613	. . 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 1636  ax10o 1809  equs5a 1870  equs5e 1871  ax11o 1892

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