Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-ax11-lem9 Structured version   Unicode version

Theorem wl-ax11-lem9 28577
Description: The easy part when  x coincides with  y. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem9  |-  ( A. x  x  =  y  ->  ( A. y A. x ph  <->  A. x A. y ph ) )

Proof of Theorem wl-ax11-lem9
StepHypRef Expression
1 biidd 237 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1 2027 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
32aecoms 2012 . . 3  |-  ( A. y  y  =  x  ->  ( A. x ph  <->  A. y ph ) )
43dral1 2027 . 2  |-  ( A. y  y  =  x  ->  ( A. y A. x ph  <->  A. x A. y ph ) )
54aecoms 2012 1  |-  ( A. x  x  =  y  ->  ( A. y A. x ph  <->  A. x A. y ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator