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Axiom ax-10 2217
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 2216 ("o" for "old") and was replaced with this shorter ax-10 2217 in May 2008. The old axiom is proved from this one as theorem ax10o 2038. Conversely, this axiom is proved from ax-10o 2216 as theorem ax10from10o 2254.

This axiom was proved redundant in July 2015. See theorem ax10 2025.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 2025. (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 1653 . . 3  wff  x  =  y
43, 1wal 1549 . 2  wff  A. x  x  =  y
52, 1weq 1653 . . 3  wff  y  =  x
65, 2wal 1549 . 2  wff  A. y 
y  =  x
74, 6wi 4 1  wff  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
This axiom is referenced by:  ax10o-o  2280
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