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Axiom ax-11 1727
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 2051). 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 2093 ("o" for "old") and was replaced with this shorter ax-11 1727 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1947. Conversely, this axiom is proved from ax-11o 2093 as theorem ax11 2107.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2093) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2049 and ax11v2 1945 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1707) 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 1633 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1530 . . 3  wff  A. y ph
63, 4wi 4 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1530 . . 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  1728  equs5a  1840  equs5e  1841  dvelimv  1892  ax10lem6  1896  a16g  1898  ax10o  1905  ax11o  1947  dvelimvNEW7  29439  ax10oNEW7  29453  ax11oNEW7  29507  a16gNEW7  29521  a12study4  29739  ax10lem17ALT  29745  a12study10  29758  a12study10n  29759  ax9lem3  29764  ax9lem17  29778
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