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Theorem prtlem100 26706
Description: Lemma for prter3 26733. (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 632 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
2 eldifsn 3929 . . . 4  |-  ( x  e.  ( A  \  { (/) } )  <->  ( x  e.  A  /\  x  =/=  (/) ) )
32anbi1i 678 . . 3  |-  ( ( x  e.  ( A 
\  { (/) } )  /\  ( B  e.  x  /\  ph )
)  <->  ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) ) )
4 ne0i 3636 . . . . . . 7  |-  ( B  e.  x  ->  x  =/=  (/) )
54pm4.71ri 616 . . . . . 6  |-  ( B  e.  x  <->  ( x  =/=  (/)  /\  B  e.  x ) )
65anbi1i 678 . . . . 5  |-  ( ( B  e.  x  /\  ph )  <->  ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) )
7 anass 632 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) 
<->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) )
86, 7bitri 242 . . . 4  |-  ( ( B  e.  x  /\  ph )  <->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph )
) )
98anbi2i 677 . . 3  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
101, 3, 93bitr4ri 271 . 2  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  ( A  \  { (/)
} )  /\  ( B  e.  x  /\  ph ) ) )
1110rexbii2 2736 1  |-  ( E. x  e.  A  ( B  e.  x  /\  ph )  <->  E. x  e.  ( A  \  { (/) } ) ( B  e.  x  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   (/)c0 3630   {csn 3816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-v 2960  df-dif 3325  df-nul 3631  df-sn 3822
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