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Theorem aecom-o 2103
Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 1899 using ax-10o 2091. Unlike ax10from10o 2129, this version does not require ax-17 1606. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Assertion
Ref Expression
aecom-o  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem aecom-o
StepHypRef Expression
1 ax-10o 2091 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
21pm2.43i 43 . 2  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
3 equcomi 1664 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1549 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 15 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530
This theorem is referenced by:  aecoms-o  2104  naecoms-o  2130  aev-o  2134  ax11indalem  2149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-10o 2091
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