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Theorem opeqex 4447
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 2609 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( <. A ,  B >.  =/=  (/)  <->  <. C ,  D >.  =/=  (/) ) )
2 opnz 4432 . 2  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
3 opnz 4432 . 2  |-  ( <. C ,  D >.  =/=  (/) 
<->  ( C  e.  _V  /\  D  e.  _V )
)
41, 2, 33bitr3g 279 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   (/)c0 3628   <.cop 3817
This theorem is referenced by:  oteqex2  4448  oteqex  4449  epelg  4495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823
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