MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv5OLD Structured version   Unicode version

Theorem reusv5OLD 4725
Description: Two ways to express single-valuedness of a class expression  C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
reusv5OLD  |-  ( B  =/=  (/)  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  x  =  C )
)
Distinct variable groups:    x, A    x, y, B    x, C
Allowed substitution hints:    A( y)    C( y)

Proof of Theorem reusv5OLD
StepHypRef Expression
1 equid 1688 . . . . 5  |-  y  =  y
21biantru 492 . . . 4  |-  ( y  e.  B  <->  ( y  e.  B  /\  y  =  y ) )
32exbii 1592 . . 3  |-  ( E. y  y  e.  B  <->  E. y ( y  e.  B  /\  y  =  y ) )
4 n0 3629 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
5 df-rex 2703 . . 3  |-  ( E. y  e.  B  y  =  y  <->  E. y
( y  e.  B  /\  y  =  y
) )
63, 4, 53bitr4i 269 . 2  |-  ( B  =/=  (/)  <->  E. y  e.  B  y  =  y )
7 reusv1 4715 . . 3  |-  ( E. y  e.  B  y  =  y  ->  ( E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C )  <->  E. x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) ) )
81a1bi 328 . . . . 5  |-  ( x  =  C  <->  ( y  =  y  ->  x  =  C ) )
98ralbii 2721 . . . 4  |-  ( A. y  e.  B  x  =  C  <->  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
109reubii 2886 . . 3  |-  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
119rexbii 2722 . . 3  |-  ( E. x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
127, 10, 113bitr4g 280 . 2  |-  ( E. y  e.  B  y  =  y  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  x  =  C ) )
136, 12sylbi 188 1  |-  ( B  =/=  (/)  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  x  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699   (/)c0 3620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-v 2950  df-dif 3315  df-nul 3621
  Copyright terms: Public domain W3C validator