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Theorem prtlem100 32493
Description: Lemma for prter3 32518. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100  |-  ( E. x  e.  A  ( B  e.  x  /\  ph )  <->  E. x  e.  ( A  \  { (/) } ) ( B  e.  x  /\  ph )
)

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 661 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
2 eldifsn 4088 . . . 4  |-  ( x  e.  ( A  \  { (/) } )  <->  ( x  e.  A  /\  x  =/=  (/) ) )
32anbi1i 709 . . 3  |-  ( ( x  e.  ( A 
\  { (/) } )  /\  ( B  e.  x  /\  ph )
)  <->  ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) ) )
4 ne0i 3728 . . . . . . 7  |-  ( B  e.  x  ->  x  =/=  (/) )
54pm4.71ri 645 . . . . . 6  |-  ( B  e.  x  <->  ( x  =/=  (/)  /\  B  e.  x ) )
65anbi1i 709 . . . . 5  |-  ( ( B  e.  x  /\  ph )  <->  ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) )
7 anass 661 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) 
<->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) )
86, 7bitri 257 . . . 4  |-  ( ( B  e.  x  /\  ph )  <->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph )
) )
98anbi2i 708 . . 3  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
101, 3, 93bitr4ri 286 . 2  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  ( A  \  { (/)
} )  /\  ( B  e.  x  /\  ph ) ) )
1110rexbii2 2879 1  |-  ( E. x  e.  A  ( B  e.  x  /\  ph )  <->  E. x  e.  ( A  \  { (/) } ) ( B  e.  x  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   (/)c0 3722   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-v 3033  df-dif 3393  df-nul 3723  df-sn 3960
This theorem is referenced by: (None)
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