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Axiom ax-9 1946
Description: Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).

We prove in ax9 1950 that this axiom can be recovered from its weakened version ax9v 1947 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 1946 should be ax9v 1947. See the comment of ax9v 1947 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 1950 instead. (New usage is discouraged.)

Assertion
Ref Expression
ax-9 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Detailed syntax breakdown of Axiom ax-9
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1822 . 2 wff 𝑥 = 𝑦
4 vz . . . 4 setvar 𝑧
54, 1wel 1938 . . 3 wff 𝑧𝑥
64, 2wel 1938 . . 3 wff 𝑧𝑦
75, 6wi 4 . 2 wff (𝑧𝑥𝑧𝑦)
83, 7wi 4 1 wff (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
This axiom is referenced by:  ax9v  1947
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