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Theorem bj-dtru 33865
Description: Remove dependency on ax-13 1968 from dtru 4644. (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 33864 . . . . 5  |-  E. w  x  e.  w
2 ax-nul 4582 . . . . . 6  |-  E. z A. x  -.  x  e.  z
3 sp 1808 . . . . . 6  |-  ( A. x  -.  x  e.  z  ->  -.  x  e.  z )
42, 3eximii 1637 . . . . 5  |-  E. z  -.  x  e.  z
5 eeanv 1957 . . . . 5  |-  ( E. w E. z ( x  e.  w  /\  -.  x  e.  z
)  <->  ( E. w  x  e.  w  /\  E. z  -.  x  e.  z ) )
61, 4, 5mpbir2an 918 . . . 4  |-  E. w E. z ( x  e.  w  /\  -.  x  e.  z )
7 ax-9 1771 . . . . . . 7  |-  ( w  =  z  ->  (
x  e.  w  ->  x  e.  z )
)
87com12 31 . . . . . 6  |-  ( x  e.  w  ->  (
w  =  z  ->  x  e.  z )
)
98con3dimp 441 . . . . 5  |-  ( ( x  e.  w  /\  -.  x  e.  z
)  ->  -.  w  =  z )
1092eximi 1636 . . . 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 1748 . . . . . . 7  |-  ( z  =  y  ->  (
w  =  z  <->  w  =  y ) )
1312notbid 294 . . . . . 6  |-  ( z  =  y  ->  ( -.  w  =  z  <->  -.  w  =  y ) )
14 ax-7 1739 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  y  ->  w  =  y )
)
1514con3d 133 . . . . . . 7  |-  ( x  =  w  ->  ( -.  w  =  y  ->  -.  x  =  y ) )
1615bj-spimevv 33766 . . . . . 6  |-  ( -.  w  =  y  ->  E. x  -.  x  =  y )
1713, 16syl6bi 228 . . . . 5  |-  ( z  =  y  ->  ( -.  w  =  z  ->  E. x  -.  x  =  y ) )
18 ax-7 1739 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1918con3d 133 . . . . . . 7  |-  ( x  =  z  ->  ( -.  z  =  y  ->  -.  x  =  y ) )
2019bj-spimevv 33766 . . . . . 6  |-  ( -.  z  =  y  ->  E. x  -.  x  =  y )
2120a1d 25 . . . . 5  |-  ( -.  z  =  y  -> 
( -.  w  =  z  ->  E. x  -.  x  =  y
) )
2217, 21pm2.61i 164 . . . 4  |-  ( -.  w  =  z  ->  E. x  -.  x  =  y )
2322exlimivv 1699 . . 3  |-  ( E. w E. z  -.  w  =  z  ->  E. x  -.  x  =  y )
2411, 23ax-mp 5 . 2  |-  E. x  -.  x  =  y
25 exnal 1628 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2624, 25mpbi 208 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-nul 4582  ax-pow 4631
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600
This theorem is referenced by:  bj-axc16b  33866  bj-eunex  33867  bj-dtrucor  33868  bj-dvdemo1  33870
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