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Axiom ax-13 1625
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 left-hand side of the  e. binary predicate. Axiom scheme C12' 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. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-13  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )

Detailed syntax breakdown of Axiom ax-13
StepHypRef Expression
1 vx . . 3  set  x
2 vy . . 3  set  y
31, 2weq 1620 . 2  wff  x  =  y
4 vz . . . 4  set  z
51, 4wel 1622 . . 3  wff  x  e.  z
62, 4wel 1622 . . 3  wff  y  e.  z
75, 6wi 6 . 2  wff  ( x  e.  z  ->  y  e.  z )
83, 7wi 6 1  wff  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
Colors of variables: wff set class
This axiom is referenced by:  elequ1  1831  el  4164  axextdfeq  23524  ax13dfeq  23525  exnel  23529  elequ1K  28239
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