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Theorem alequcom-o 2093
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 alequcom 1889 using ax-10o 2082. Unlike ax10from10o 2118, this version does not require ax-17 1608. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Assertion
Ref Expression
alequcom-o  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem alequcom-o
StepHypRef Expression
1 ax-10o 2082 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
21pm2.43i 45 . 2  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
3 equcomi 1650 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1551 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 17 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  alequcoms-o  2094  nalequcoms-o  2119  aev-o  2123  ax11indalem  2137
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-10o 2082
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