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Theorem reusv5OLD 4560
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 1662 . . . . 5  |-  y  =  y
21biantru 491 . . . 4  |-  ( y  e.  B  <->  ( y  e.  B  /\  y  =  y ) )
32exbii 1572 . . 3  |-  ( E. y  y  e.  B  <->  E. y ( y  e.  B  /\  y  =  y ) )
4 n0 3477 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
5 df-rex 2562 . . 3  |-  ( E. y  e.  B  y  =  y  <->  E. y
( y  e.  B  /\  y  =  y
) )
63, 4, 53bitr4i 268 . 2  |-  ( B  =/=  (/)  <->  E. y  e.  B  y  =  y )
7 reusv1 4550 . . 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 327 . . . . 5  |-  ( x  =  C  <->  ( y  =  y  ->  x  =  C ) )
98ralbii 2580 . . . 4  |-  ( A. y  e.  B  x  =  C  <->  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
109reubii 2739 . . 3  |-  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
119rexbii 2581 . . 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 279 . 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 187 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-v 2803  df-dif 3168  df-nul 3469
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