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Theorem euop2 4266
Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
Hypothesis
Ref Expression
euop2.1  |-  A  e. 
_V
Assertion
Ref Expression
euop2  |-  ( E! x E. y ( x  =  <. A , 
y >.  /\  ph )  <->  E! y ph )
Distinct variable groups:    ph, x    x, A    x, y
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem euop2
StepHypRef Expression
1 opex 4237 . 2  |-  <. A , 
y >.  e.  _V
2 euop2.1 . . 3  |-  A  e. 
_V
32moop2 4261 . 2  |-  E* y  x  =  <. A , 
y >.
41, 3euxfr2 2950 1  |-  ( E! x E. y ( x  =  <. A , 
y >.  /\  ph )  <->  E! y ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788   <.cop 3643
This theorem is referenced by:  dfac5lem1  7750
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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