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Theorem otth2 4407
Description: Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
otth.1  |-  A  e. 
_V
otth.2  |-  B  e. 
_V
otth.3  |-  R  e. 
_V
Assertion
Ref Expression
otth2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )

Proof of Theorem otth2
StepHypRef Expression
1 otth.1 . . . 4  |-  A  e. 
_V
2 otth.2 . . . 4  |-  B  e. 
_V
31, 2opth 4403 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
43anbi1i 677 . 2  |-  ( (
<. A ,  B >.  = 
<. C ,  D >.  /\  R  =  S )  <-> 
( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
5 opex 4395 . . 3  |-  <. A ,  B >.  e.  _V
6 otth.3 . . 3  |-  R  e. 
_V
75, 6opth 4403 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( <. A ,  B >.  =  <. C ,  D >.  /\  R  =  S ) )
8 df-3an 938 . 2  |-  ( ( A  =  C  /\  B  =  D  /\  R  =  S )  <->  ( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
94, 7, 83bitr4i 269 1  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785
This theorem is referenced by:  otth  4408  oprabid  6072  eloprabga  6127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791
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