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Theorem opeqex 3986
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2101 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( x  e.  <. A ,  B >.  <->  x  e.  <. C ,  D >. ) )
21exbidv 1706 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( E. x  x  e.  <. A ,  B >.  <->  E. x  x  e.  <. C ,  D >. ) )
3 opm 3971 . 2  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
4 opm 3971 . 2  |-  ( E. x  x  e.  <. C ,  D >.  <->  ( C  e.  _V  /\  D  e. 
_V ) )
52, 3, 43bitr3g 211 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    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557   <.cop 3378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384
This theorem is referenced by:  epelg  4027
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