MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-11 Unicode version

Axiom ax-11 1715
Description: Axiom of 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 2038). 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 2080 ("o" for "old") and was replaced with this shorter ax-11 1715 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1934. Conversely, this axiom is proved from ax-11o 2080 as theorem ax11 2094.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2080) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2036 and ax11v2 1932 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1695) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)

Assertion
Ref Expression
ax-11  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Detailed syntax breakdown of Axiom ax-11
StepHypRef Expression
1 vx . . 3  set  x
2 vy . . 3  set  y
31, 2weq 1624 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1527 . . 3  wff  A. y ph
63, 4wi 4 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1527 . . 3  wff  A. x
( x  =  y  ->  ph )
85, 7wi 4 . 2  wff  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) )
93, 8wi 4 1  wff  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
This axiom is referenced by:  sp  1716  equs5a  1828  equs5e  1829  dvelimv  1879  ax10lem6  1883  a16g  1885  ax10o  1892  ax11o  1934  a12study4  29117  ax10lem17ALT  29123  a12study10  29136  a12study10n  29137  ax9lem3  29142  ax9lem17  29156
  Copyright terms: Public domain W3C validator