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Axiom ax-c9 32170
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This axiom is obsolete and should no longer be used.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 axc9 2102. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c9  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )

Detailed syntax breakdown of Axiom ax-c9
StepHypRef Expression
1 vz . . . . 5  setvar  z
2 vx . . . . 5  setvar  x
31, 2weq 1783 . . . 4  wff  z  =  x
43, 1wal 1435 . . 3  wff  A. z 
z  =  x
54wn 3 . 2  wff  -.  A. z  z  =  x
6 vy . . . . . 6  setvar  y
71, 6weq 1783 . . . . 5  wff  z  =  y
87, 1wal 1435 . . . 4  wff  A. z 
z  =  y
98wn 3 . . 3  wff  -.  A. z  z  =  y
102, 6weq 1783 . . . 4  wff  x  =  y
1110, 1wal 1435 . . . 4  wff  A. z  x  =  y
1210, 11wi 4 . . 3  wff  ( x  =  y  ->  A. z  x  =  y )
139, 12wi 4 . 2  wff  ( -. 
A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
)
145, 13wi 4 1  wff  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff setvar class
This axiom is referenced by:  hbae-o  32181  ax13fromc9  32184  equid1  32186  hbequid  32188  equid1ALT  32204  dvelimf-o  32208  ax5eq  32211
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