MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  naecoms Structured version   Unicode version

Theorem naecoms 2081
Description: A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
Hypothesis
Ref Expression
naecoms.1  |-  ( -. 
A. x  x  =  y  ->  ph )
Assertion
Ref Expression
naecoms  |-  ( -. 
A. y  y  =  x  ->  ph )

Proof of Theorem naecoms
StepHypRef Expression
1 aecom 2079 . 2  |-  ( A. x  x  =  y  <->  A. y  y  =  x )
2 naecoms.1 . 2  |-  ( -. 
A. x  x  =  y  ->  ph )
31, 2sylnbir 307 1  |-  ( -. 
A. y  y  =  x  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-12 1880  ax-13 2028
This theorem depends on definitions:  df-bi 187  df-an 371  df-ex 1636  df-nf 1640
This theorem is referenced by:  sb9  2195  eujustALT  2243  nfcvf2  2592  axpowndlem2  9007  wl-sbcom2d  31391  wl-mo2dnae  31399
  Copyright terms: Public domain W3C validator