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Axiom ax-11 1486
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 1866). 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 1807, ax11v2 1800 and ax-11o 1803. (Contributed by NM, 5-Aug-1993.)

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

Detailed syntax breakdown of Axiom ax-11
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1483 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1333 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1333 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
85, 7wi 4 . 2 wff (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
93, 8wi 4 1 wff (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
This axiom is referenced by:  ax10o  1695  equs5a  1774  sbcof2  1790  ax11o  1802  ax11v  1807
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