Theorem otth2 4303

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 4299	. . 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 4291	. . 3 |- <. A , B >. e. _V
6		otth.3	. . 3 |- R e. _V
7	5, 6	opth 4299	. 2 |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( <. A , B >. = <. C , D >. /\ R = S ) )
8		df-3an 983	. 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 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   <.cop 3646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  otth  4304  oprabid  5999  eloprabga  6055