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Theorem reusv5OLD 4647
Description: TODO-NM: What shall be done with this OLD theorem? 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 1796 . . . . 5  |-  y  =  y
21biantru 503 . . . 4  |-  ( y  e.  B  <->  ( y  e.  B  /\  y  =  y ) )
32exbii 1672 . . 3  |-  ( E. y  y  e.  B  <->  E. y ( y  e.  B  /\  y  =  y ) )
4 n0 3793 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
5 df-rex 2810 . . 3  |-  ( E. y  e.  B  y  =  y  <->  E. y
( y  e.  B  /\  y  =  y
) )
63, 4, 53bitr4i 277 . 2  |-  ( B  =/=  (/)  <->  E. y  e.  B  y  =  y )
7 reusv1 4637 . . 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 335 . . . . 5  |-  ( x  =  C  <->  ( y  =  y  ->  x  =  C ) )
98ralbii 2885 . . . 4  |-  ( A. y  e.  B  x  =  C  <->  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
109reubii 3041 . . 3  |-  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
119rexbii 2956 . . 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 288 . 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 195 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   E!wreu 2806   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by: (None)
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