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Axiom ax-11 1761
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 2174). 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 2217 ("o" for "old") and was replaced with this shorter ax-11 1761 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 2077. Conversely, this axiom is proved from ax-11o 2217 as theorem ax11 2231.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2217) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2171 and ax11v2 2074 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1736) 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 1653 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1549 . . 3  wff  A. y ph
63, 4wi 4 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1549 . . 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:  19.8a  1762  spOLD  1764  equs5a  1909  equs5e  1910  equs5eOLD  1911  ax10o2  2024  dvelimvOLD  2028  ax10oOLD  2039  a16gOLD  2049  ax11o  2077  ax11vALT  2172  dvelimvNEW7  29399  ax10oNEW7  29413  ax11oNEW7  29469  a16gNEW7  29483
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