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Axiom ax-7 1776
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 1782). 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, 10-Jan-1993.)

Assertion
Ref Expression
ax-7  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 vx . . 3  setvar  x
2 vy . . 3  setvar  y
31, 2weq 1720 . 2  wff  x  =  y
4 vz . . . 4  setvar  z
51, 4weq 1720 . . 3  wff  x  =  z
62, 4weq 1720 . . 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 setvar class
This axiom is referenced by:  equid  1777  equcomi  1779  equtr  1782  equequ1  1784  cbvaev  1803  aev  1929  aevOLD  2048  aevALT  2049  axc16i  2050  equveli  2074  equveliOLD  2075  hbequid  2225  equidqe  2238  aev-o  2247  mo3OLD  2310  axext3  2423  dtru  4628  axextnd  8969  2spotmdisj  25046  wl-aetr  29959  wl-exeq  29963  wl-aleq  29964  wl-nfeqfb  29966  ax6e2eq  33198  ax6e2eqVD  33575  bj-aev  34205  bj-dtru  34266
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