Theorem otthg 2785

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 2777	. . . 4 |- <.A, B>. e. V
2		otthg.3	. . . 4 |- R e. V
3	1, 2	opthg 2783	. . 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 2783	. . . 4 |- (D e. F -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))
7	6	anbi1d 616	. . 3 |- (D e. F -> ((<.A, B>. = <.C, D>. /\ R = S) <-> ((A = C /\ B = D) /\ R = S)))
8	3, 7	sylan9bbr 540	. 2 |- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> ((A = C /\ B = D) /\ R = S)))
9		df-3an 776	. 2 |- ((A = C /\ B = D /\ R = S) <-> ((A = C /\ B = D) /\ R = S))
10	8, 9	syl6bbr 537	1 |- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> (A = C /\ B = D /\ R = S)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  Vcvv 1807  <.cop 2407

This theorem is referenced by:  eloprabg 3998  elo 10381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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