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Theorem opthg2 4246
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 4245 . 2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
2 eqcom 2286 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  <. C ,  D >.  =  <. A ,  B >. )
3 eqcom 2286 . . 3  |-  ( A  =  C  <->  C  =  A )
4 eqcom 2286 . . 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 1623    e. wcel 1685   <.cop 3644
This theorem is referenced by:  opth2  4247  fliftel  5770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650
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