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Axiom ax-11 1331
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 1674). 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.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1618, ax11v2 1611 and ax-11o 1614. (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 1326 . 2 wff x = y
4 wph . . . 4 wff φ
54, 2wal 1267 . . 3 wff yφ
63, 4wi 4 . . . 4 wff (x = yφ)
76, 1wal 1267 . . 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  1516  equs5a  1585  sbcof2  1601  ax11o  1613  ax11v  1618
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