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Theorem opth2 4567
Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
Hypotheses
Ref Expression
opth2.1  |-  C  e. 
_V
opth2.2  |-  D  e. 
_V
Assertion
Ref Expression
opth2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)

Proof of Theorem opth2
StepHypRef Expression
1 opth2.1 . 2  |-  C  e. 
_V
2 opth2.2 . 2  |-  D  e. 
_V
3 opthg2 4566 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
41, 2, 3mp2an 667 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881
This theorem is referenced by:  eqvinop  4572  opelxp  4865  fsn  5878  opiota  6632  canthwe  8814  ltresr  9303  fmucndlem  19825  mat1dimelbas  30750  diblsmopel  34538  cdlemn7  34570  dihordlem7  34581  xihopellsmN  34621  dihopellsm  34622  dihpN  34703
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