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Axiom ax-8 1683
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 1690). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

Note that this axiom is still valid even when any two or all three of  x,  y, and  z are replaced with the same variable since they do not have any distinct variable (Metamath's $d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (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  set  x
2 vy . . 3  set  y
31, 2weq 1650 . 2  wff  x  =  y
4 vz . . . 4  set  z
51, 4weq 1650 . . 3  wff  x  =  z
62, 4weq 1650 . . 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:  equid  1684  equidOLD  1685  equcomi  1687  equtr  1690  equequ1  1692  equequ1OLD  1693  ax12olem1  1972  ax12olem2  1973  ax12olem1OLD  1977  ax10lem1  1988  aev  2013  equvini  2042  equviniOLD  2043  equveli  2044  equveliOLD  2045  ax16i  2103  hbequid  2218  equidqe  2231  aev-o  2240  mo  2284  dtru  4358  axextnd  8430  2spotmdisj  28179  a9e2eq  28363  a9e2eqVD  28737  aevwAUX7  29238  equviniNEW7  29243  equveliNEW7  29244  ax16iNEW7  29267  ax7w9AUX7  29372  alcomw9bAUX7  29373
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