MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-12o Unicode version

Axiom ax-12o 2094
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This axiom is obsolete and should no longer be used.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 1887. (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 1633 . . . 4  wff  z  =  x
43, 1wal 1530 . . 3  wff  A. z 
z  =  x
54wn 3 . 2  wff  -.  A. z  z  =  x
6 vy . . . . . 6  set  y
71, 6weq 1633 . . . . 5  wff  z  =  y
87, 1wal 1530 . . . 4  wff  A. z 
z  =  y
98wn 3 . . 3  wff  -.  A. z  z  =  y
102, 6weq 1633 . . . 4  wff  x  =  y
1110, 1wal 1530 . . . 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 set class
This axiom is referenced by:  hbae-o  2105  ax12from12o  2108  equid1  2110  hbequid  2112  equid1ALT  2128  dvelimf-o  2132  ax17eq  2135
  Copyright terms: Public domain W3C validator