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Theorem exopxfr 6369
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 4976 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y  e.  _V  E. z  e. 
_V  ps )
3 rexv 2930 . 2  |-  ( E. y  e.  _V  E. z  e.  _V  ps  <->  E. y E. z  e.  _V  ps )
4 rexv 2930 . . 3  |-  ( E. z  e.  _V  ps  <->  E. z ps )
54exbii 1589 . 2  |-  ( E. y E. z  e. 
_V  ps  <->  E. y E. z ps )
62, 3, 53bitri 263 1  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y E. z ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    = wceq 1649   E.wrex 2667   _Vcvv 2916   <.cop 3777    X. cxp 4835
This theorem is referenced by:  exopxfr2  6370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-iun 4055  df-opab 4227  df-xp 4843  df-rel 4844
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