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Theorem reusv5OLD 4544
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 1644 . . . . 5  |-  y  =  y
21biantru 491 . . . 4  |-  ( y  e.  B  <->  ( y  e.  B  /\  y  =  y ) )
32exbii 1569 . . 3  |-  ( E. y  y  e.  B  <->  E. y ( y  e.  B  /\  y  =  y ) )
4 n0 3464 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
5 df-rex 2549 . . 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 4534 . . 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 2567 . . . 4  |-  ( A. y  e.  B  x  =  C  <->  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
109reubii 2726 . . 3  |-  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  ( y  =  y  ->  x  =  C ) )
119rexbii 2568 . . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   (/)c0 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-v 2790  df-dif 3155  df-nul 3456
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