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Theorem aecom 2145
Description: Commutation law for identical variable specifiers. Both sides of the biconditional are true when  x and  y are substituted with the same variable. (Contributed by NM, 10-May-1993.) Changed to a biconditional. (Revised by BJ, 26-Sep-2019.)
Assertion
Ref Expression
aecom  |-  ( A. x  x  =  y  <->  A. y  y  =  x )

Proof of Theorem aecom
StepHypRef Expression
1 axc11n 2143 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 axc11n 2143 . 2  |-  ( A. y  y  =  x  ->  A. x  x  =  y )
31, 2impbii 191 1  |-  ( A. x  x  =  y  <->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  aecoms  2146  naecoms  2147  wl-nfae1  31860
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