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Theorem exopxfr2 6184
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
Hypothesis
Ref Expression
exopxfr2.1  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
exopxfr2  |-  ( Rel 
A  ->  ( E. x  e.  A  ph  <->  E. y E. z ( <. y ,  z >.  e.  A  /\  ps ) ) )
Distinct variable groups:    x, y,
z, A    ph, y, z    ps, x
Allowed substitution hints:    ph( x)    ps( y, z)

Proof of Theorem exopxfr2
StepHypRef Expression
1 df-rel 4696 . . . . . . 7  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
21biimpi 186 . . . . . 6  |-  ( Rel 
A  ->  A  C_  ( _V  X.  _V ) )
32sseld 3179 . . . . 5  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
43adantrd 454 . . . 4  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  ph )  ->  x  e.  ( _V  X.  _V )
) )
54pm4.71rd 616 . . 3  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  ph )  <->  ( x  e.  ( _V  X.  _V )  /\  ( x  e.  A  /\  ph )
) ) )
65rexbidv2 2566 . 2  |-  ( Rel 
A  ->  ( E. x  e.  A  ph  <->  E. x  e.  ( _V  X.  _V ) ( x  e.  A  /\  ph )
) )
7 eleq1 2343 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  <->  <. y ,  z
>.  e.  A ) )
8 exopxfr2.1 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
97, 8anbi12d 691 . . 3  |-  ( x  =  <. y ,  z
>.  ->  ( ( x  e.  A  /\  ph ) 
<->  ( <. y ,  z
>.  e.  A  /\  ps ) ) )
109exopxfr 6183 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ( x  e.  A  /\  ph ) 
<->  E. y E. z
( <. y ,  z
>.  e.  A  /\  ps ) )
116, 10syl6bb 252 1  |-  ( Rel 
A  ->  ( E. x  e.  A  ph  <->  E. y E. z ( <. y ,  z >.  e.  A  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   Rel wrel 4694
This theorem is referenced by:  dvhopellsm  31307
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  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  df-iun 3907  df-opab 4078  df-xp 4695  df-rel 4696
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