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Axiom ax-11 1394
 Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀x(x = y → φ) is a way of expressing "y substituted for x in wff φ " (cf. sb6 1763). 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 1705, ax11v2 1698 and ax-11o 1701. (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 1389 . 2 wff x = y
4 wph . . . 4 wff φ
54, 2wal 1240 . . 3 wff yφ
63, 4wi 4 . . . 4 wff (x = yφ)
76, 1wal 1240 . . 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  1600  equs5a  1672  sbcof2  1688  ax11o  1700  ax11v  1705
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