Theorem otth2 4226

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 4222	. . 3 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
4	3	anbi1i 455	. 2 |- ( ( <. A , B >. = <. C , D >. /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
5	1, 2	opex 4214	. . 3 |- <. A , B >. e. _V
6		otth.3	. . 3 |- R e. _V
7	5, 6	opth 4222	. 2 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( <. A , B >. = <. C , D >. /\ R = S ) )
8		df-3an 975	. 2 |- ( ( A = C /\ B = D /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
9	4, 7, 8	3bitr4i 211	1 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( A = C /\ B = D /\ R = S ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   <.cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  otth  4227  oprabid  5885  eloprabga  5940