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Theorem prtlem100 30762
Description: Lemma for prter3 30789. (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 647 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
2 eldifsn 4069 . . . 4  |-  ( x  e.  ( A  \  { (/) } )  <->  ( x  e.  A  /\  x  =/=  (/) ) )
32anbi1i 693 . . 3  |-  ( ( x  e.  ( A 
\  { (/) } )  /\  ( B  e.  x  /\  ph )
)  <->  ( ( x  e.  A  /\  x  =/=  (/) )  /\  ( B  e.  x  /\  ph ) ) )
4 ne0i 3717 . . . . . . 7  |-  ( B  e.  x  ->  x  =/=  (/) )
54pm4.71ri 631 . . . . . 6  |-  ( B  e.  x  <->  ( x  =/=  (/)  /\  B  e.  x ) )
65anbi1i 693 . . . . 5  |-  ( ( B  e.  x  /\  ph )  <->  ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) )
7 anass 647 . . . . 5  |-  ( ( ( x  =/=  (/)  /\  B  e.  x )  /\  ph ) 
<->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) )
86, 7bitri 249 . . . 4  |-  ( ( B  e.  x  /\  ph )  <->  ( x  =/=  (/)  /\  ( B  e.  x  /\  ph )
) )
98anbi2i 692 . . 3  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  A  /\  (
x  =/=  (/)  /\  ( B  e.  x  /\  ph ) ) ) )
101, 3, 93bitr4ri 278 . 2  |-  ( ( x  e.  A  /\  ( B  e.  x  /\  ph ) )  <->  ( x  e.  ( A  \  { (/)
} )  /\  ( B  e.  x  /\  ph ) ) )
1110rexbii2 2882 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 184    /\ wa 367    e. wcel 1826    =/= wne 2577   E.wrex 2733    \ cdif 3386   (/)c0 3711   {csn 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rex 2738  df-v 3036  df-dif 3392  df-nul 3712  df-sn 3945
This theorem is referenced by: (None)
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