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Axiom ax-11 1624
Description: Axiom of Variable 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 1992). 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 1940 ("o" for "old") and was replaced with this shorter ax-11 1624 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1939. Conversely, this axiom is proved from ax-11o 1940 as theorem ax11 1941.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1940) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

Interestingly, if the wff expression substituted for  ph contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1940 (from which the ax-11 1624 instance follows by theorem ax11 1941.) The proof is by induction on formula length, using ax11eq 2105 and ax11el 2106 for the basis steps and ax11indn 2108, ax11indi 2109, and ax11inda 2113 for the induction steps. (This paragraph is true provided we use ax-10o 1835 in place of ax-10 1678.)

See also ax11v 1990 and ax11v2 1935 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. (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 1620 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1532 . . 3  wff  A. y ph
63, 4wi 6 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1532 . . 3  wff  A. x
( x  =  y  ->  ph )
85, 7wi 6 . 2  wff  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) )
93, 8wi 6 1  wff  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
This axiom is referenced by:  ax12o10lem3  1637  ax10lem23  1672  ax10lem24  1673  a16gALT  1679  ax4  1691  ax10o  1834  equs5a  1911  equs5e  1912  ax11o  1939  a12study4  27806  ax10lem17ALT  27812  a12study10  27825  a12study10n  27826  ax9lem3  27831  ax9lem17  27845
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