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Theorem prtlem100 32423
Description: Lemma for prter3 32448. (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 654 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
2 eldifsn 4096 . . . 4  |-  ( x  e.  ( A  \  { (/) } )  <->  ( x  e.  A  /\  x  =/=  (/) ) )
32anbi1i 700 . . 3  |-  ( ( x  e.  ( A 
\  { (/) } )  /\  ( B  e.  x  /\  ph )
)  <->  ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) ) )
4 ne0i 3736 . . . . . . 7  |-  ( B  e.  x  ->  x  =/=  (/) )
54pm4.71ri 638 . . . . . 6  |-  ( B  e.  x  <->  ( x  =/=  (/)  /\  B  e.  x ) )
65anbi1i 700 . . . . 5  |-  ( ( B  e.  x  /\  ph )  <->  ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) )
7 anass 654 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) 
<->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) )
86, 7bitri 253 . . . 4  |-  ( ( B  e.  x  /\  ph )  <->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph )
) )
98anbi2i 699 . . 3  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
101, 3, 93bitr4ri 282 . 2  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  ( A  \  { (/)
} )  /\  ( B  e.  x  /\  ph ) ) )
1110rexbii2 2886 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 188    /\ wa 371    e. wcel 1886    =/= wne 2621   E.wrex 2737    \ cdif 3400   (/)c0 3730   {csn 3967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-rex 2742  df-v 3046  df-dif 3406  df-nul 3731  df-sn 3968
This theorem is referenced by: (None)
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