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

Theorem bj-axsep 31378
Description: Remove dependency on ax-13 2057 from axsep 4545. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axsep  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Distinct variable groups:    x, y,
z    ph, y, z
Allowed substitution hint:    ph( x)

Proof of Theorem bj-axsep
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1755 . . . 4  |-  F/ y ( w  =  x  /\  ph )
21bj-axrep5 31377 . . 3  |-  ( A. w ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )  ->  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) ) )
3 equtr 1850 . . . . . . . 8  |-  ( y  =  w  ->  (
w  =  x  -> 
y  =  x ) )
4 equcomi 1847 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
53, 4syl6 34 . . . . . . 7  |-  ( y  =  w  ->  (
w  =  x  ->  x  =  y )
)
65adantrd 469 . . . . . 6  |-  ( y  =  w  ->  (
( w  =  x  /\  ph )  ->  x  =  y )
)
76alrimiv 1767 . . . . 5  |-  ( y  =  w  ->  A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
87a1d 26 . . . 4  |-  ( y  =  w  ->  (
w  e.  z  ->  A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) ) )
98bj-spimevv 31283 . . 3  |-  ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )
102, 9mpg 1665 . 2  |-  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
11 an12 804 . . . . . . 7  |-  ( ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
1211exbii 1712 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )
13 elequ1 1875 . . . . . . . 8  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1413anbi1d 709 . . . . . . 7  |-  ( w  =  x  ->  (
( w  e.  z  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
1514bj-equsexvv 31312 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1612, 15bitr3i 254 . . . . 5  |-  ( E. w ( w  e.  z  /\  ( w  =  x  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1716bibi2i 314 . . . 4  |-  ( ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
1817albii 1685 . . 3  |-  ( A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) )
1918exbii 1712 . 2  |-  ( E. y A. x ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
2010, 19mpbi 211 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657
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-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-rep 4536
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator