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Axiom ax-7 1883
Description: Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 1897). 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 C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 1892 that this axiom can be recovered from its weakened version ax7v 1884 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 1883 should be ax7v 1884. See the comment of ax7v 1884 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1892 instead. (New usage is discouraged.)

Assertion
Ref Expression
ax-7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1822 . 2 wff 𝑥 = 𝑦
4 vz . . . 4 setvar 𝑧
51, 4weq 1822 . . 3 wff 𝑥 = 𝑧
62, 4weq 1822 . . 3 wff 𝑦 = 𝑧
75, 6wi 4 . 2 wff (𝑥 = 𝑧𝑦 = 𝑧)
83, 7wi 4 1 wff (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
This axiom is referenced by:  ax7v  1884
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