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

Theorem bj-el 31456
Description: Remove dependency on ax-13 2102 from el 4599. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-el  |-  E. y  x  e.  y
Distinct variable group:    x, y

Proof of Theorem bj-el
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bj-zfpow 31455 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax9 1911 . . . . 5  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1784 . . . 4  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax8 1904 . . . 4  |-  ( z  =  x  ->  (
z  e.  y  ->  x  e.  y )
)
53, 4embantd 56 . . 3  |-  ( z  =  x  ->  (
( A. y ( y  e.  z  -> 
y  e.  x )  ->  z  e.  y )  ->  x  e.  y ) )
65bj-spimvv 31367 . 2  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
71, 6eximii 1720 1  |-  E. y  x  e.  y
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-pow 4595
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  bj-dtru  31457  bj-dvdemo2  31463
  Copyright terms: Public domain W3C validator