Axiom ax-12 1768

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 2192). 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 2251 and was replaced with this shorter ax-12 1768 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2089. Conversely, this axiom is proved from ax-c15 2251 as theorem ax12 2265.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 2251) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 2189 and ax12v2 2086 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1743) 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 1671	. 2
4		wph	. . . 4
5	4, 2	wal 1564	. . 3
6	3, 4	wi 4	. . . 4
7	6, 1	wal 1564	. . 3
8	5, 7	wi 4	. 2
9	3, 8	wi 4	1

This axiom is referenced by:  19.8a  1769  equs5a  1913  equs5e  1914  equs5eOLD  1915  axc112  2029  dvelimvOLD  2033  axc11OLD  2044  a16gOLD  2055  axc15  2089  ax12vALT  2190  dvelimvNEW11  31014  axc11NEW11  31028  axc15NEW11  31084  a16gNEW11  31098