Theorem otth2 3996

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

Step	Hyp	Ref	Expression
1		otth.1	. . . 4 |- A e. _V
2		otth.2	. . . 4 |- B e. _V
3	1, 2	opth 3992	. . 3 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
4	3	anbi1i 445	. 2 |- ( ( <. A , B >. = <. C , D >. /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
5	1, 2	opex 3984	. . 3 |- <. A , B >. e. _V
6		otth.3	. . 3 |- R e. _V
7	5, 6	opth 3992	. 2 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( <. A , B >. = <. C , D >. /\ R = S ) )
8		df-3an 921	. 2 |- ( ( A = C /\ B = D /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
9	4, 7, 8	3bitr4i 210	1 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( A = C /\ B = D /\ R = S ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  otth  3997  oprabid  5557  eloprabga  5611