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Theorem opthg2 4326
Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem opthg2
StepHypRef Expression
1 opthg 4325 . 2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
2 eqcom 2360 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  <. C ,  D >.  =  <. A ,  B >. )
3 eqcom 2360 . . 3  |-  ( A  =  C  <->  C  =  A )
4 eqcom 2360 . . 3  |-  ( B  =  D  <->  D  =  B )
53, 4anbi12i 678 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( C  =  A  /\  D  =  B )
)
61, 2, 53bitr4g 279 1  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   <.cop 3719
This theorem is referenced by:  opth2  4327  fliftel  5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725
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