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Axiom ax-11 1757
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 2148). 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 2191 ("o" for "old") and was replaced with this shorter ax-11 1757 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 2047. Conversely, this axiom is proved from ax-11o 2191 as theorem ax11 2205.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2191) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2145 and ax11v2 2045 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1732) 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 1650 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1546 . . 3  wff  A. y ph
63, 4wi 4 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1546 . . 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  1758  spOLD  1760  equs5a  1905  equs5e  1906  equs5eOLD  1907  ax10lem3  1990  dvelimvOLD  1994  ax10o  2001  a16gOLD  2013  ax11o  2047  ax11vALT  2146  axi11e  2382  dvelimvNEW7  29168  ax10oNEW7  29182  ax11oNEW7  29236  a16gNEW7  29250
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