MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexdifsn Structured version   Unicode version

Theorem rexdifsn 4088
Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
Assertion
Ref Expression
rexdifsn  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )

Proof of Theorem rexdifsn
StepHypRef Expression
1 eldifsn 4084 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21anbi1i 695 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( (
x  e.  A  /\  x  =/=  B )  /\  ph ) )
3 anass 649 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
42, 3bitri 249 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
54rexbii2 2825 1  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1757    =/= wne 2641   E.wrex 2793    \ cdif 3409   {csn 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-rex 2798  df-v 3056  df-dif 3415  df-sn 3962
This theorem is referenced by:  symgfix2  16009  usgra2pth0  30426  2spot2iun2spont  30534  dihatexv  35265  lcfl8b  35431
  Copyright terms: Public domain W3C validator