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 1480). 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 1570 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 1570 and ax-17 1349 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 1570 and ax-17 1349 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  1355  equidqeOLD  1356  equid  1475  equcomi  1477  equtr  1480  equequ1  1483  equvini  1517  equveli  1518  aev  1568  ax16i  1610  mo  1782  ax4  1915  a12lem1  1932  a12study  1934  a12study3  1937
  Copyright terms: Public domain W3C validator