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Theorem exopxfr2 6411
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 4885 . . . . . . 7  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
21biimpi 187 . . . . . 6  |-  ( Rel 
A  ->  A  C_  ( _V  X.  _V ) )
32sseld 3347 . . . . 5  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
43adantrd 455 . . . 4  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  ph )  ->  x  e.  ( _V  X.  _V )
) )
54pm4.71rd 617 . . 3  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  ph )  <->  ( x  e.  ( _V  X.  _V )  /\  ( x  e.  A  /\  ph )
) ) )
65rexbidv2 2728 . 2  |-  ( Rel 
A  ->  ( E. x  e.  A  ph  <->  E. x  e.  ( _V  X.  _V ) ( x  e.  A  /\  ph )
) )
7 eleq1 2496 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  <->  <. y ,  z
>.  e.  A ) )
8 exopxfr2.1 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
97, 8anbi12d 692 . . 3  |-  ( x  =  <. y ,  z
>.  ->  ( ( x  e.  A  /\  ph ) 
<->  ( <. y ,  z
>.  e.  A  /\  ps ) ) )
109exopxfr 6410 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ( x  e.  A  /\  ph ) 
<->  E. y E. z
( <. y ,  z
>.  e.  A  /\  ps ) )
116, 10syl6bb 253 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956    C_ wss 3320   <.cop 3817    X. cxp 4876   Rel wrel 4883
This theorem is referenced by:  dvhopellsm  31915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-iun 4095  df-opab 4267  df-xp 4884  df-rel 4885
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