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Theorem bj-dtru 31478
Description: Remove dependency on ax-13 2104 from dtru 4592. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dtru  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem bj-dtru
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-el 31477 . . . . 5  |-  E. w  x  e.  w
2 ax-nul 4527 . . . . . 6  |-  E. z A. x  -.  x  e.  z
3 sp 1957 . . . . . 6  |-  ( A. x  -.  x  e.  z  ->  -.  x  e.  z )
42, 3eximii 1717 . . . . 5  |-  E. z  -.  x  e.  z
5 eeanv 2093 . . . . 5  |-  ( E. w E. z ( x  e.  w  /\  -.  x  e.  z
)  <->  ( E. w  x  e.  w  /\  E. z  -.  x  e.  z ) )
61, 4, 5mpbir2an 934 . . . 4  |-  E. w E. z ( x  e.  w  /\  -.  x  e.  z )
7 ax9 1917 . . . . . . 7  |-  ( w  =  z  ->  (
x  e.  w  ->  x  e.  z )
)
87com12 31 . . . . . 6  |-  ( x  e.  w  ->  (
w  =  z  ->  x  e.  z )
)
98con3dimp 448 . . . . 5  |-  ( ( x  e.  w  /\  -.  x  e.  z
)  ->  -.  w  =  z )
1092eximi 1716 . . . 4  |-  ( E. w E. z ( x  e.  w  /\  -.  x  e.  z
)  ->  E. w E. z  -.  w  =  z )
116, 10ax-mp 5 . . 3  |-  E. w E. z  -.  w  =  z
12 equequ2 1876 . . . . . . 7  |-  ( z  =  y  ->  (
w  =  z  <->  w  =  y ) )
1312notbid 301 . . . . . 6  |-  ( z  =  y  ->  ( -.  w  =  z  <->  -.  w  =  y ) )
14 ax7 1868 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  y  ->  w  =  y )
)
1514con3d 140 . . . . . . 7  |-  ( x  =  w  ->  ( -.  w  =  y  ->  -.  x  =  y ) )
1615bj-spimevv 31389 . . . . . 6  |-  ( -.  w  =  y  ->  E. x  -.  x  =  y )
1713, 16syl6bi 236 . . . . 5  |-  ( z  =  y  ->  ( -.  w  =  z  ->  E. x  -.  x  =  y ) )
18 ax7 1868 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1918con3d 140 . . . . . . 7  |-  ( x  =  z  ->  ( -.  z  =  y  ->  -.  x  =  y ) )
2019bj-spimevv 31389 . . . . . 6  |-  ( -.  z  =  y  ->  E. x  -.  x  =  y )
2120a1d 25 . . . . 5  |-  ( -.  z  =  y  -> 
( -.  w  =  z  ->  E. x  -.  x  =  y
) )
2217, 21pm2.61i 169 . . . 4  |-  ( -.  w  =  z  ->  E. x  -.  x  =  y )
2322exlimivv 1786 . . 3  |-  ( E. w E. z  -.  w  =  z  ->  E. x  -.  x  =  y )
2411, 23ax-mp 5 . 2  |-  E. x  -.  x  =  y
25 exnal 1707 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2624, 25mpbi 213 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376   A.wal 1450   E.wex 1671
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-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-nul 4527  ax-pow 4579
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676
This theorem is referenced by:  bj-axc16b  31479  bj-eunex  31480  bj-dtrucor  31481  bj-dvdemo1  31483
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