ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeqex Unicode version

Theorem opeqex 4050
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 2148 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( x  e.  <. A ,  B >.  <->  x  e.  <. C ,  D >. ) )
21exbidv 1750 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( E. x  x  e.  <. A ,  B >.  <->  E. x  x  e.  <. C ,  D >. ) )
3 opm 4035 . 2  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
4 opm 4035 . 2  |-  ( E. x  x  e.  <. C ,  D >.  <->  ( C  e.  _V  /\  D  e. 
_V ) )
52, 3, 43bitr3g 220 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 102    <-> wb 103    = wceq 1287   E.wex 1424    e. wcel 1436   _Vcvv 2615   <.cop 3434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440
This theorem is referenced by:  epelg  4091
  Copyright terms: Public domain W3C validator