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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.

The antecedents imply  -.  x  =  z as well as  -.  x  =  y, so  x must be a different variable from both  y and  z if the antecedents hold. The axiom essentially states that if  x is neither  y nor  z, then it is irrelevant to the truth of  y  =  z.

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 1881.

Although this version is shorter, the original version ax-12o 1664 may be more intuitive to understand, as well as 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 use only ax12v 1634 for further derivation. Thus ax12v 1634 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1634 or ax12 1881. (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
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