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Axiom ax-8 1436
Description: Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1637). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1436 through ax-16 1737 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1737 and ax-17 1460 are still valid even when 𝑥, 𝑦, and 𝑧 are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1737 and ax-17 1460 only. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Detailed syntax breakdown of Axiom ax-8
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1433 . 2 wff 𝑥 = 𝑦
4 vz . . . 4 setvar 𝑧
51, 4weq 1433 . . 3 wff 𝑥 = 𝑧
62, 4weq 1433 . . 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 set class
This axiom is referenced by:  hbequid  1447  equidqe  1466  equid  1630  equcomi  1633  equtr  1637  equequ1  1640  equvini  1683  equveli  1684  aev  1735  ax16i  1781  mo23  1984
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