Axiom ax-12 1794

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 2142). 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 2200 and was replaced with this shorter ax-12 1794 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2045. Conversely, this axiom is proved from ax-c15 2200 as theorem ax12 2214.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 2200) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 1795 and ax12v2 2043 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1769) 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 1696	. 2
4		wph	. . . 4
5	4, 2	wal 1368	. . 3
6	3, 4	wi 4	. . . 4
7	6, 1	wal 1368	. . 3
8	5, 7	wi 4	. 2
9	3, 8	wi 4	1

This axiom is referenced by:  ax12v  1795  ax12vOLD  1796  19.8aOLD  1798  axc112  1875  equs5a  1918  equs5e  1919  axc15  2045  mo2vOLD  2270  mo2vOLDOLD  2271  bj-axc15v  33110