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Theorem wl-ax11-lem9 31987
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 245 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1 2174 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
32aecoms 2161 . . 3  |-  ( A. y  y  =  x  ->  ( A. x ph  <->  A. y ph ) )
43dral1 2174 . 2  |-  ( A. y  y  =  x  ->  ( A. y A. x ph  <->  A. x A. y ph ) )
54aecoms 2161 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 189   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
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