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Theorem naecoms 2026
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 2024 . 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 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  sb9  2149  eujustALT  2278  nfcvf2  2655  axpowndlem2  8974  wl-sbcom2d  29864  wl-mo2dnae  29872
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