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Axiom ax-14 1626
Description: Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the  e. binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-14  |-  ( x  =  y  ->  (
z  e.  x  -> 
z  e.  y ) )

Detailed syntax breakdown of Axiom ax-14
StepHypRef Expression
1 vx . . 3  set  x
2 vy . . 3  set  y
31, 2weq 1620 . 2  wff  x  =  y
4 vz . . . 4  set  z
54, 1wel 1622 . . 3  wff  z  e.  x
64, 2wel 1622 . . 3  wff  z  e.  y
75, 6wi 6 . 2  wff  ( z  e.  x  ->  z  e.  y )
83, 7wi 6 1  wff  ( x  =  y  ->  (
z  e.  x  -> 
z  e.  y ) )
Colors of variables: wff set class
This axiom is referenced by:  elequ2  1832  el  4164  dtru  4173  fv3  5474  elirrv  7279  ax10ext  26974  elequ2K  28240
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