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Axiom ax-10 2084
Description: Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 2083 ("o" for "old") and was replaced with this shorter ax-10 2084 in May 2008. The old axiom is proved from this one as theorem ax10o 1896. Conversely, this axiom is proved from ax-10o 2083 as theorem ax10from10o 2120.

This axiom was proved redundant in July 2015. See theorem ax10 1888.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1888. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Assertion
Ref Expression
ax-10  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Detailed syntax breakdown of Axiom ax-10
StepHypRef Expression
1 vx . . . 4  set  x
2 vy . . . 4  set  y
31, 2weq 1626 . . 3  wff  x  =  y
43, 1wal 1529 . 2  wff  A. x  x  =  y
52, 1weq 1626 . . 3  wff  y  =  x
65, 2wal 1529 . 2  wff  A. y 
y  =  x
74, 6wi 6 1  wff  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
This axiom is referenced by:  ax10o-o  2145
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