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Axiom ax-12 1633
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x y  =  z ) ) ). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent  -.  x  =  y to hold,  x and  y must have different values and thus cannot be the same object-language variable. Similarly,  x and  z cannot be the same object-language variable. Therefore,  x will not occur in the wff 
y  =  z when the first two antecedents hold, so analogous to ax-17 1628, the conclusion  ( y  =  z  ->  A. x
y  =  z ) follows.

To simplify the above form of the axiom, we exploit the  y  =  z antecedent to show that  -.  x  =  y is equivalent to  -.  x  =  z, so one of them is redundant and can be discarded as ax12b 1834 shows.

The original version of this axiom was ax-12o 1664 ("o" for "old") and was replaced with this shorter ax-12 1633 in December 2015. The old axiom is proved from this one as theorem ax12o 1663. Conversely, this axiom is proved from ax-12o 1664 as theorem ax12 1882.

Although this version is shorter, the original version ax-12o 1664 may be more practical to work with because of the "distinctor" form of its antecedents.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs  x ,  z and  y ,  z. To show that, we add these restrictions to theorem ax12v 1634 and use only ax12v 1634 for further derivations. Thus ax12v 1634 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1634 or ax-12o 1664. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Assertion
Ref Expression
ax-12  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Detailed syntax breakdown of Axiom ax-12
StepHypRef Expression
1 vx . . . 4  set  x
2 vy . . . 4  set  y
31, 2weq 1620 . . 3  wff  x  =  y
43wn 5 . 2  wff  -.  x  =  y
5 vz . . . 4  set  z
62, 5weq 1620 . . 3  wff  y  =  z
76, 1wal 1532 . . 3  wff  A. x  y  =  z
86, 7wi 6 . 2  wff  ( y  =  z  ->  A. x  y  =  z )
94, 8wi 6 1  wff  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
This axiom is referenced by:  ax12v  1634  ax12vX  28281
  Copyright terms: Public domain W3C validator