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Theorem wl-ax11-lem2 31880
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem2  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  A. x  u  =  y )
Distinct variable group:    x, u

Proof of Theorem wl-ax11-lem2
StepHypRef Expression
1 sp 1914 . . 3  |-  ( A. u  u  =  y  ->  u  =  y )
2 aev 2003 . . . 4  |-  ( A. x  x  =  u  ->  A. x  x  =  y )
3 pm2.21 111 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. x  x  =  u ) )
42, 3impbid2 207 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( A. x  x  =  u  <->  A. x  x  =  y ) )
51, 4anim12i 568 . 2  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( u  =  y  /\  ( A. x  x  =  u 
<-> 
A. x  x  =  y ) ) )
6 wl-aleq 31832 . 2  |-  ( A. x  u  =  y  <->  ( u  =  y  /\  ( A. x  x  =  u  <->  A. x  x  =  y ) ) )
75, 6sylibr 215 1  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  A. x  u  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  wl-ax11-lem3  31881
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