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Theorem opeqex 4139
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 2179 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( x  e.  <. A ,  B >.  <->  x  e.  <. C ,  D >. ) )
21exbidv 1779 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( E. x  x  e.  <. A ,  B >.  <->  E. x  x  e.  <. C ,  D >. ) )
3 opm 4124 . 2  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
4 opm 4124 . 2  |-  ( E. x  x  e.  <. C ,  D >.  <->  ( C  e.  _V  /\  D  e. 
_V ) )
52, 3, 43bitr3g 221 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 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   <.cop 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504
This theorem is referenced by:  epelg  4180
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