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Axiom ax-8 1643
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 1652). 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 1624 . 2  wff  x  =  y
4 vz . . . 4  set  z
51, 4weq 1624 . . 3  wff  x  =  z
62, 4weq 1624 . . 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  1644  equcomi  1646  equequ1  1648  equtr  1652  ax12olem1  1868  ax10lem1  1876  equvini  1927  equveli  1928  aev  1931  ax16i  1986  hbequid  2099  equidqe  2112  aev-o  2121  mo  2165  dtru  4201  axextnd  8213  a9e2eq  28323  a9e2eqVD  28683  a12lem1  29130  a12study10  29136  a12study10n  29137  ax9lem1  29140  ax9lem2  29141  ax9lem6  29145  ax9lem14  29153  ax9lem15  29154  ax9vax9  29158
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