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Theorem bj-eunex 30949
Description: Remove dependency on ax-13 2026 from eunex 4587. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-eunex  |-  ( E! x ph  ->  E. x  -.  ph )

Proof of Theorem bj-eunex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bj-dtru 30947 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1653 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 178 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1743 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 464 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 eu3v 2268 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
7 exnal 1669 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
85, 6, 73imtr4i 266 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1403   E.wex 1633   E!weu 2238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-nul 4525  ax-pow 4572
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243
This theorem is referenced by: (None)
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