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Axiom ax-15 2177
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 1623; see theorem ax15 2054. Alternately, ax-17 1623 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1623. We retain ax-15 2177 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1623, 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This axiom is obsolete and should no longer be used.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 2054. (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 1650 . . . 4  wff  z  =  x
43, 1wal 1546 . . 3  wff  A. z 
z  =  x
54wn 3 . 2  wff  -.  A. z  z  =  x
6 vy . . . . . 6  set  y
71, 6weq 1650 . . . . 5  wff  z  =  y
87, 1wal 1546 . . . 4  wff  A. z 
z  =  y
98wn 3 . . 3  wff  -.  A. z  z  =  y
102, 6wel 1718 . . . 4  wff  x  e.  y
1110, 1wal 1546 . . . 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  2223  ax11el  2228
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