Theorem otth2 4285

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 4281	. . 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 4273	. . 3 |- <. A , B >. e. _V
6		otth.3	. . 3 |- R e. _V
7	5, 6	opth 4281	. 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 2176   _Vcvv 2772   <.cop 3636

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  otth  4286  oprabid  5976  eloprabga  6032