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Theorem reusv5OLD 4674
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 1683 . . . . 5  |-  y  =  y
21biantru 492 . . . 4  |-  ( y  e.  B  <->  ( y  e.  B  /\  y  =  y ) )
32exbii 1589 . . 3  |-  ( E. y  y  e.  B  <->  E. y ( y  e.  B  /\  y  =  y ) )
4 n0 3581 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
5 df-rex 2656 . . 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 4664 . . 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 2674 . . . 4  |-  ( A. y  e.  B  x  =  C  <->  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
109reubii 2838 . . 3  |-  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
119rexbii 2675 . . 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 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   E!wreu 2652   (/)c0 3572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-v 2902  df-dif 3267  df-nul 3573
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