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Theorem euop2 4459
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 4430 . 2  |-  <. A , 
y >.  e.  _V
2 euop2.1 . . 3  |-  A  e. 
_V
32moop2 4454 . 2  |-  E* y  x  =  <. A , 
y >.
41, 3euxfr2 3121 1  |-  ( E! x E. y ( x  =  <. A , 
y >.  /\  ph )  <->  E! y ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283   _Vcvv 2958   <.cop 3819
This theorem is referenced by:  dfac5lem1  8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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