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Axiom ax-8 1392
 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 1592). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. Axioms ax-8 1392 through ax-16 1692 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 1692 and ax-17 1416 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 1692 and ax-17 1416 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 1389 . 2 wff x = y
4 vz . . . 4 setvar z
51, 4weq 1389 . . 3 wff x = z
62, 4weq 1389 . . 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  1403  equidqe  1422  equid  1586  equcomi  1589  equtr  1592  equequ1  1595  equvini  1638  equveli  1639  aev  1690  ax16i  1735  mo23  1938
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