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Theorem otth2 4123
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 4119 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
43anbi1i 451 . 2  |-  ( (
<. A ,  B >.  = 
<. C ,  D >.  /\  R  =  S )  <-> 
( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
51, 2opex 4111 . . 3  |-  <. A ,  B >.  e.  _V
6 otth.3 . . 3  |-  R  e. 
_V
75, 6opth 4119 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( <. A ,  B >.  =  <. C ,  D >.  /\  R  =  S ) )
8 df-3an 947 . 2  |-  ( ( A  =  C  /\  B  =  D  /\  R  =  S )  <->  ( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
94, 7, 83bitr4i 211 1  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   _Vcvv 2657   <.cop 3496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502
This theorem is referenced by:  otth  4124  oprabid  5757  eloprabga  5812
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