Theorem otthg 1900

Description: Ordered triple theorem.

Hypotheses

Ref	Expression
otthg.1	|- A e. V
otthg.2	|- B e. V
otthg.3	|- R e. V

Assertion

Ref	Expression
otthg	|- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> (A = C /\ B = D /\ R = S)))

Proof of Theorem otthg

Step	Hyp	Ref	Expression
1		opex 1893	. . . 4 |- <.A, B>. e. V
2		otthg.3	. . . 4 |- R e. V
3	1, 2	opthg 1899	. . 3 |- (S e. G -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> (<.A, B>. = <.C, D>. /\ R = S)))
4		otthg.1	. . . . 5 |- A e. V
5		otthg.2	. . . . 5 |- B e. V
6	4, 5	opthg 1899	. . . 4 |- (D e. F -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))
7	6	anbi1d 469	. . 3 |- (D e. F -> ((<.A, B>. = <.C, D>. /\ R = S) <-> ((A = C /\ B = D) /\ R = S)))
8	3, 7	sylan9bbr 419	. 2 |- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> ((A = C /\ B = D) /\ R = S)))
9		df-3an 583	. 2 |- ((A = C /\ B = D /\ R = S) <-> ((A = C /\ B = D) /\ R = S))
10	8, 9	syl6bbr 416	1 |- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> (A = C /\ B = D /\ R = S)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810

This theorem is referenced by:  eloprabg 3035

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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