ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-8 GIF version

Axiom ax-8 1328
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 1485). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1328 through ax-16 1581 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 1581 and ax-17 1350 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 1581 and ax-17 1350 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 1325 . 2 wff x = y
4 vz . . . 4 set z
51, 4weq 1325 . . 3 wff x = z
62, 4weq 1325 . . 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  1339  equidqe  1356  equidqeOLD  1357  equid  1480  equcomi  1482  equtr  1485  equequ1  1488  equvini  1525  equveli  1526  aev  1579  ax16i  1621  mo  2091
  Copyright terms: Public domain W3C validator