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Axiom ax-11 1305
Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent x(x = yφ) is a way of expressing "y substituted for x in wff φ " (cf. sb6 1581). 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.

In classical logic, ax-11o 1530 can be derived from this axiom, as can be seen at ax11o 1529. However, the current proof of ax11o 1529 is not valid intuitionistically.

In classical logic, this axiom is metalogically independent from the others, but not logically independent. Lack of logical independence means that if the wff expression substituted for φ contains no wff variables, the resulting statement can be proved without invoking this axiom. The current proofs of this are not valid in intuitionistic logic, however. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1530 (from which the ax-11 1305 instance follows by theorem ax11 1531.) The proof is by induction on formula length, using ax11eq 1786 and ax11el 1787 for the basis steps and ax11indn 1788, ax11indi 1789, and ax11inda 1793 for the induction steps. Many of those theorems rely on classical logic for their proofs.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1579, ax11v2 1527 and ax-11o 1530. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-11 (x = y → (yφx(x = yφ)))

Detailed syntax breakdown of Axiom ax-11
StepHypRef Expression
1 vx . . 3 set x
2 vy . . 3 set y
31, 2weq 1300 . 2 wff x = y
4 wph . . . 4 wff φ
54, 2wal 1251 . . 3 wff yφ
63, 4wi 4 . . . 4 wff (x = yφ)
76, 1wal 1251 . . 3 wff x(x = yφ)
85, 7wi 4 . 2 wff (yφx(x = yφ))
93, 8wi 4 1 wff (x = y → (yφx(x = yφ)))
Colors of variables: wff set class
This axiom is referenced by:  ax10o  1451  equs5a  1507  ax11o  1529  ax4  1784
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