Theorem otthg 2796

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 2788	. . . 4 |- <.A, B>. e. V
2		otthg.3	. . . 4 |- R e. V
3	1, 2	opthg 2794	. . 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 2794	. . . 4 |- (D e. F -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))
7	6	anbi1d 619	. . 3 |- (D e. F -> ((<.A, B>. = <.C, D>. /\ R = S) <-> ((A = C /\ B = D) /\ R = S)))
8	3, 7	sylan9bbr 543	. 2 |- ((D e. F /\ S e. G) -> (<.<.A, B>., R>. = <.<.C, D>., S>. <-> ((A = C /\ B = D) /\ R = S)))
9		df-3an 779	. 2 |- ((A = C /\ B = D /\ R = S) <-> ((A = C /\ B = D) /\ R = S))
10	8, 9	syl6bbr 540	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 777   = wceq 958   e. wcel 960  Vcvv 1814  <.cop 2415

This theorem is referenced by:  eloprabg 4013  elo 10439

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420

Copyright terms: Public domain