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Axiom ax-15 2087
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-17 1605; see theorem ax15 1968. Alternately, ax-17 1605 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1605. We retain ax-15 2087 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1605, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax15 1968. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-15  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )

Detailed syntax breakdown of Axiom ax-15
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, 6wel 1688 . . . 4  wff  x  e.  y
1110, 1wal 1529 . . . 4  wff  A. z  x  e.  y
1210, 11wi 6 . . 3  wff  ( x  e.  y  ->  A. z  x  e.  y )
139, 12wi 6 . 2  wff  ( -. 
A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
)
145, 13wi 6 1  wff  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )
Colors of variables: wff set class
This axiom is referenced by:  ax17el  2132  ax11el  2136
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