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Axiom ax-11 1374
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 1744). 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 1686, ax11v2 1679 and ax-11o 1682. (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 setvar x
2 vy . . 3 setvar y
31, 2weq 1369 . 2 wff x = y
4 wph . . . 4 wff φ
54, 2wal 1224 . . 3 wff yφ
63, 4wi 4 . . . 4 wff (x = yφ)
76, 1wal 1224 . . 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  1581  equs5a  1653  sbcof2  1669  ax11o  1681  ax11v  1686
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