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Axiom ax-8 1283
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 1418). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1283 through ax-16 1502 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 1502 and ax-17 1297 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 1502 and ax-17 1297 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 set x
2 vy . . 3 set y
31, 2weq 1280 . 2 wff x = y
4 vz . . . 4 set z
51, 4weq 1280 . . 3 wff x = z
62, 4weq 1280 . . 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  1293  equidqe  1301  equidqeOLD  1302  equid  1413  equcomi  1415  equtr  1418  equequ1  1421  equvini  1455  equveli  1456  aev  1500  ax16i  1563  mo  1704  ax4  1809  a12lem1  1823  a12study  1825  a12study3  1828
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