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Axiom ax-c9 2214
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This axiom is obsolete and should no longer be used.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 2019. (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 1705 . . . 4  wff  z  =  x
43, 1wal 1377 . . 3  wff  A. z 
z  =  x
54wn 3 . 2  wff  -.  A. z  z  =  x
6 vy . . . . . 6  setvar  y
71, 6weq 1705 . . . . 5  wff  z  =  y
87, 1wal 1377 . . . 4  wff  A. z 
z  =  y
98wn 3 . . 3  wff  -.  A. z  z  =  y
102, 6weq 1705 . . . 4  wff  x  =  y
1110, 1wal 1377 . . . 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  2225  ax13fromc9  2228  equid1  2230  hbequid  2232  equid1ALT  2248  dvelimf-o  2252  ax5eq  2255
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