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Axiom ax-8 1411
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 1611). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1411 through ax-16 1711 are the axioms having to do with equality, substitution, and logical properties of our binary predicate  e. (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1711 and ax-17 1435 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 1711 and ax-17 1435 only. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-8  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )

Detailed syntax breakdown of Axiom ax-8
StepHypRef Expression
1 vx . . 3  setvar  x
2 vy . . 3  setvar  y
31, 2weq 1408 . 2  wff  x  =  y
4 vz . . . 4  setvar  z
51, 4weq 1408 . . 3  wff  x  =  z
62, 4weq 1408 . . 3  wff  y  =  z
75, 6wi 4 . 2  wff  ( x  =  z  ->  y  =  z )
83, 7wi 4 1  wff  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
Colors of variables: wff set class
This axiom is referenced by:  hbequid  1422  equidqe  1441  equid  1605  equcomi  1608  equtr  1611  equequ1  1614  equvini  1657  equveli  1658  aev  1709  ax16i  1754  mo23  1957
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