Axiom ax-12 1854

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 2173). 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 2220 and was replaced with this shorter ax-12 1854 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2085. Conversely, this axiom is proved from ax-c15 2220 as theorem ax12 2234.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 2220) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 1855 and ax12v2 2083 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1829) 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 1733	. 2
4		wph	. . . 4
5	4, 2	wal 1393	. . 3
6	3, 4	wi 4	. . . 4
7	6, 1	wal 1393	. . 3
8	5, 7	wi 4	. 2
9	3, 8	wi 4	1

This axiom is referenced by:  ax12v  1855  ax12vOLD  1856  19.8aOLD  1858  axc112  1937  equs5a  1978  equs5e  1979  axc15  2085  mo2vOLD  2290  mo2vOLDOLD  2291  bj-axc15v  34330