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Theorem exopxfr 6413
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 5020 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y  e.  _V  E. z  e. 
_V  ps )
3 rexv 2972 . 2  |-  ( E. y  e.  _V  E. z  e.  _V  ps  <->  E. y E. z  e.  _V  ps )
4 rexv 2972 . . 3  |-  ( E. z  e.  _V  ps  <->  E. z ps )
54exbii 1593 . 2  |-  ( E. y E. z  e. 
_V  ps  <->  E. y E. z ps )
62, 3, 53bitri 264 1  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y E. z ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   E.wex 1551    = wceq 1653   E.wrex 2708   _Vcvv 2958   <.cop 3819    X. cxp 4879
This theorem is referenced by:  exopxfr2  6414
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  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  df-iun 4097  df-opab 4270  df-xp 4887  df-rel 4888
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