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Axiom ax-12o 2086
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This axiom is obsolete and should no longer be used. It is proved above as theorem ax12o 1879. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-12o  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )

Detailed syntax breakdown of Axiom ax-12o
StepHypRef Expression
1 vz . . . . 5  set  z
2 vx . . . . 5  set  x
31, 2weq 1626 . . . 4  wff  z  =  x
43, 1wal 1529 . . 3  wff  A. z 
z  =  x
54wn 5 . 2  wff  -.  A. z  z  =  x
6 vy . . . . . 6  set  y
71, 6weq 1626 . . . . 5  wff  z  =  y
87, 1wal 1529 . . . 4  wff  A. z 
z  =  y
98wn 5 . . 3  wff  -.  A. z  z  =  y
102, 6weq 1626 . . . 4  wff  x  =  y
1110, 1wal 1529 . . . 4  wff  A. z  x  =  y
1210, 11wi 6 . . 3  wff  ( x  =  y  ->  A. z  x  =  y )
139, 12wi 6 . 2  wff  ( -. 
A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
)
145, 13wi 6 1  wff  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff set class
This axiom is referenced by:  hbae-o  2096  ax12  2099  equid1  2101  hbequid  2103  equid1ALT  2119  dvelimf-o  2123  ax17eq  2126
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