Theorem otth2 4243

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 4239	. . 3 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
4	3	anbi1i 458	. 2 |- ( ( <. A , B >. = <. C , D >. /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
5	1, 2	opex 4231	. . 3 |- <. A , B >. e. _V
6		otth.3	. . 3 |- R e. _V
7	5, 6	opth 4239	. 2 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( <. A , B >. = <. C , D >. /\ R = S ) )
8		df-3an 980	. 2 |- ( ( A = C /\ B = D /\ R = S ) <-> ( ( A = C /\ B = D ) /\ R = S ) )
9	4, 7, 8	3bitr4i 212	1 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( A = C /\ B = D /\ R = S ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2739   <.cop 3597

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603

This theorem is referenced by:  otth  4244  oprabid  5909  eloprabga  5964