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Theorem prtlem100 26721
Description: Lemma for prter3 26750. (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 630 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
2 eldifsn 3749 . . . 4  |-  ( x  e.  ( A  \  { (/) } )  <->  ( x  e.  A  /\  x  =/=  (/) ) )
32anbi1i 676 . . 3  |-  ( ( x  e.  ( A 
\  { (/) } )  /\  ( B  e.  x  /\  ph )
)  <->  ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) ) )
4 ne0i 3461 . . . . . . 7  |-  ( B  e.  x  ->  x  =/=  (/) )
54pm4.71ri 614 . . . . . 6  |-  ( B  e.  x  <->  ( x  =/=  (/)  /\  B  e.  x ) )
65anbi1i 676 . . . . 5  |-  ( ( B  e.  x  /\  ph )  <->  ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) )
7 anass 630 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) 
<->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) )
86, 7bitri 240 . . . 4  |-  ( ( B  e.  x  /\  ph )  <->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph )
) )
98anbi2i 675 . . 3  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
101, 3, 93bitr4ri 269 . 2  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  ( A  \  { (/)
} )  /\  ( B  e.  x  /\  ph ) ) )
1110rexbii2 2572 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 176    /\ wa 358    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456  df-sn 3646
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