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Theorem exopxfr 6183
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
exopxfr.1  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
exopxfr  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y E. z ps )
Distinct variable groups:    y, z, ph    ps, x    x, y,
z
Allowed substitution hints:    ph( x)    ps( y, z)

Proof of Theorem exopxfr
StepHypRef Expression
1 exopxfr.1 . . 3  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
21rexxp 4828 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y  e.  _V  E. z  e. 
_V  ps )
3 rexv 2802 . 2  |-  ( E. y  e.  _V  E. z  e.  _V  ps  <->  E. y E. z  e.  _V  ps )
4 rexv 2802 . . 3  |-  ( E. z  e.  _V  ps  <->  E. z ps )
54exbii 1569 . 2  |-  ( E. y E. z  e. 
_V  ps  <->  E. y E. z ps )
62, 3, 53bitri 262 1  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y E. z ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528    = wceq 1623   E.wrex 2544   _Vcvv 2788   <.cop 3643    X. cxp 4687
This theorem is referenced by:  exopxfr2  6184
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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