Theorem otth 4332

Description: Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (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
otth	|- ( <. A , B , R >. = <. C , D , S >. <-> ( A = C /\ B = D /\ R = S ) )

Proof of Theorem otth

Step	Hyp	Ref	Expression
1		df-ot 3677	. . 3 |- <. A , B , R >. = <. <. A , B >. , R >.
2		df-ot 3677	. . 3 |- <. C , D , S >. = <. <. C , D >. , S >.
3	1, 2	eqeq12i 2243	. 2 |- ( <. A , B , R >. = <. C , D , S >. <-> <. <. A , B >. , R >. = <. <. C , D >. , S >. )
4		otth.1	. . 3 |- A e. _V
5		otth.2	. . 3 |- B e. _V
6		otth.3	. . 3 |- R e. _V
7	4, 5, 6	otth2 4331	. 2 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( A = C /\ B = D /\ R = S ) )
8	3, 7	bitri 184	1 |- ( <. A , B , R >. = <. C , D , S >. <-> ( A = C /\ B = D /\ R = S ) )

Syntax hints:   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2800   <.cop 3670   <.cotp 3671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-ot 3677

This theorem is referenced by:  euotd  4345