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Theorem opeqex 4257
Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
opeqex  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( ( A  e. 
_V  /\  B  e.  _V )  <->  ( C  e. 
_V  /\  D  e.  _V ) ) )

Proof of Theorem opeqex
StepHypRef Expression
1 neeq1 2454 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( <. A ,  B >.  =/=  (/)  <->  <. C ,  D >.  =/=  (/) ) )
2 opnz 4242 . 2  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
3 opnz 4242 . 2  |-  ( <. C ,  D >.  =/=  (/) 
<->  ( C  e.  _V  /\  D  e.  _V )
)
41, 2, 33bitr3g 278 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( ( A  e. 
_V  /\  B  e.  _V )  <->  ( C  e. 
_V  /\  D  e.  _V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   <.cop 3643
This theorem is referenced by:  oteqex2  4258  oteqex  4259  epelg  4306
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-v 2790  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
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