Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-dtru Structured version   Unicode version

Theorem bj-dtru 31163
Description: Remove dependency on ax-13 2055 from dtru 4616. (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 31162 . . . . 5  |-  E. w  x  e.  w
2 ax-nul 4556 . . . . . 6  |-  E. z A. x  -.  x  e.  z
3 sp 1912 . . . . . 6  |-  ( A. x  -.  x  e.  z  ->  -.  x  e.  z )
42, 3eximii 1705 . . . . 5  |-  E. z  -.  x  e.  z
5 eeanv 2045 . . . . 5  |-  ( E. w E. z ( x  e.  w  /\  -.  x  e.  z
)  <->  ( E. w  x  e.  w  /\  E. z  -.  x  e.  z ) )
61, 4, 5mpbir2an 928 . . . 4  |-  E. w E. z ( x  e.  w  /\  -.  x  e.  z )
7 ax-9 1874 . . . . . . 7  |-  ( w  =  z  ->  (
x  e.  w  ->  x  e.  z )
)
87com12 32 . . . . . 6  |-  ( x  e.  w  ->  (
w  =  z  ->  x  e.  z )
)
98con3dimp 442 . . . . 5  |-  ( ( x  e.  w  /\  -.  x  e.  z
)  ->  -.  w  =  z )
1092eximi 1704 . . . 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 1851 . . . . . . 7  |-  ( z  =  y  ->  (
w  =  z  <->  w  =  y ) )
1312notbid 295 . . . . . 6  |-  ( z  =  y  ->  ( -.  w  =  z  <->  -.  w  =  y ) )
14 ax-7 1841 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  y  ->  w  =  y )
)
1514con3d 138 . . . . . . 7  |-  ( x  =  w  ->  ( -.  w  =  y  ->  -.  x  =  y ) )
1615bj-spimevv 31063 . . . . . 6  |-  ( -.  w  =  y  ->  E. x  -.  x  =  y )
1713, 16syl6bi 231 . . . . 5  |-  ( z  =  y  ->  ( -.  w  =  z  ->  E. x  -.  x  =  y ) )
18 ax-7 1841 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1918con3d 138 . . . . . . 7  |-  ( x  =  z  ->  ( -.  z  =  y  ->  -.  x  =  y ) )
2019bj-spimevv 31063 . . . . . 6  |-  ( -.  z  =  y  ->  E. x  -.  x  =  y )
2120a1d 26 . . . . 5  |-  ( -.  z  =  y  -> 
( -.  w  =  z  ->  E. x  -.  x  =  y
) )
2217, 21pm2.61i 167 . . . 4  |-  ( -.  w  =  z  ->  E. x  -.  x  =  y )
2322exlimivv 1770 . . 3  |-  ( E. w E. z  -.  w  =  z  ->  E. x  -.  x  =  y )
2411, 23ax-mp 5 . 2  |-  E. x  -.  x  =  y
25 exnal 1695 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2624, 25mpbi 211 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-nul 4556  ax-pow 4603
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664
This theorem is referenced by:  bj-axc16b  31164  bj-eunex  31165  bj-dtrucor  31166  bj-dvdemo1  31168
  Copyright terms: Public domain W3C validator