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Axiom ax-i12 1414
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever  z is distinct from  x and  y, and  x  =  y is true, then  x  =  y quantified with  z is also true. In other words,  z is irrelevant to the truth of 
x  =  y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom has been modified from the original ax-12 1418 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Assertion
Ref Expression
ax-i12  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )

Detailed syntax breakdown of Axiom ax-i12
StepHypRef Expression
1 vz . . . 4  setvar  z
2 vx . . . 4  setvar  x
31, 2weq 1408 . . 3  wff  z  =  x
43, 1wal 1257 . 2  wff  A. z 
z  =  x
5 vy . . . . 5  setvar  y
61, 5weq 1408 . . . 4  wff  z  =  y
76, 1wal 1257 . . 3  wff  A. z 
z  =  y
82, 5weq 1408 . . . . 5  wff  x  =  y
98, 1wal 1257 . . . . 5  wff  A. z  x  =  y
108, 9wi 4 . . . 4  wff  ( x  =  y  ->  A. z  x  =  y )
1110, 1wal 1257 . . 3  wff  A. z
( x  =  y  ->  A. z  x  =  y )
127, 11wo 639 . 2  wff  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
)
134, 12wo 639 1  wff  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1418  ax12or  1419  dveeq2  1712  dveeq2or  1713  dvelimALT  1902  dvelimfv  1903
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