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Axiom ax-15 2095
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 1606; see theorem ax15 1974. Alternately, ax-17 1606 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1606. We retain ax-15 2095 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1606, 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This axiom is obsolete and should no longer be used.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 1974. (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 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, 6wel 1697 . . . 4  wff  x  e.  y
1110, 1wal 1530 . . . 4  wff  A. z  x  e.  y
1210, 11wi 4 . . 3  wff  ( x  e.  y  ->  A. z  x  e.  y )
139, 12wi 4 . 2  wff  ( -. 
A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
)
145, 13wi 4 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  2141  ax11el  2146
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