MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exopxfr Unicode version

Theorem exopxfr 6310
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 4931 . 2  |-  ( E. x  e.  ( _V 
X.  _V ) ph  <->  E. y  e.  _V  E. z  e. 
_V  ps )
3 rexv 2887 . 2  |-  ( E. y  e.  _V  E. z  e.  _V  ps  <->  E. y E. z  e.  _V  ps )
4 rexv 2887 . . 3  |-  ( E. z  e.  _V  ps  <->  E. z ps )
54exbii 1587 . 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 1546    = wceq 1647   E.wrex 2629   _Vcvv 2873   <.cop 3732    X. cxp 4790
This theorem is referenced by:  exopxfr2  6311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-iun 4009  df-opab 4180  df-xp 4798  df-rel 4799
  Copyright terms: Public domain W3C validator