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Theorem aecom-o 2227
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 2035 using ax-10o 2215. Unlike ax10from10o 2253, this version does not require ax-17 1626. (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 2215 . . 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 1691 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1568 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 16 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549
This theorem is referenced by:  aecoms-o  2228  naecoms-o  2254  aev-o  2258  ax11indalem  2273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-10o 2215
This theorem depends on definitions:  df-bi 178  df-ex 1551
  Copyright terms: Public domain W3C validator