Axiom ax-12 1785

Description: Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent is a way of expressing " substituted for in wff " (cf. sb6 2125). 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-c15 2182 and was replaced with this shorter ax-12 1785 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2026. Conversely, this axiom is proved from ax-c15 2182 as theorem ax12 2196.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 2182) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 2122 and ax12v2 2024 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1760) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)

Assertion

Ref	Expression
ax-12

Detailed syntax breakdown of Axiom ax-12

Step	Hyp	Ref	Expression
1		vx	. . 3
2		vy	. . 3
3	1, 2	weq 1688	. 2
4		wph	. . . 4
5	4, 2	wal 1580	. . 3
6	3, 4	wi 4	. . . 4
7	6, 1	wal 1580	. . 3
8	5, 7	wi 4	. 2
9	3, 8	wi 4	1

This axiom is referenced by:  19.8a  1786  axc112  1860  equs5a  1896  equs5e  1897  axc15  2026  ax12vALT  2123  mo2v  2250  bj-axc15v  31771