Theorem opthgg 2789

Description: Ordered pair theorem. C is not required to be a set under our specific ordered pair definition.

Assertion

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

Proof of Theorem opthgg

Step	Hyp	Ref	Expression
1		opeq1 2487	. . . . . 6 |- (x = A -> <.x, y>. = <.A, y>.)
2	1	eqeq1d 1483	. . . . 5 |- (x = A -> (<.x, y>. = <.C, D>. <-> <.A, y>. = <.C, D>.))
3		eqeq1 1481	. . . . . 6 |- (x = A -> (x = C <-> A = C))
4	3	anbi1d 617	. . . . 5 |- (x = A -> ((x = C /\ y = D) <-> (A = C /\ y = D)))
5	2, 4	bibi12d 629	. . . 4 |- (x = A -> ((<.x, y>. = <.C, D>. <-> (x = C /\ y = D)) <-> (<.A, y>. = <.C, D>. <-> (A = C /\ y = D))))
6	5	imbi2d 612	. . 3 |- (x = A -> ((D e. T -> (<.x, y>. = <.C, D>. <-> (x = C /\ y = D))) <-> (D e. T -> (<.A, y>. = <.C, D>. <-> (A = C /\ y = D)))))
7		opeq2 2488	. . . . . 6 |- (y = B -> <.A, y>. = <.A, B>.)
8	7	eqeq1d 1483	. . . . 5 |- (y = B -> (<.A, y>. = <.C, D>. <-> <.A, B>. = <.C, D>.))
9		eqeq1 1481	. . . . . 6 |- (y = B -> (y = D <-> B = D))
10	9	anbi2d 616	. . . . 5 |- (y = B -> ((A = C /\ y = D) <-> (A = C /\ B = D)))
11	8, 10	bibi12d 629	. . . 4 |- (y = B -> ((<.A, y>. = <.C, D>. <-> (A = C /\ y = D)) <-> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))))
12	11	imbi2d 612	. . 3 |- (y = B -> ((D e. T -> (<.A, y>. = <.C, D>. <-> (A = C /\ y = D))) <-> (D e. T -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))))
13		visset 1813	. . . 4 |- x e. V
14		visset 1813	. . . 4 |- y e. V
15	13, 14	opthg 2788	. . 3 |- (D e. T -> (<.x, y>. = <.C, D>. <-> (x = C /\ y = D)))
16	6, 12, 15	vtocl2g 1850	. 2 |- ((A e. R /\ B e. S) -> (D e. T -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))))
17	16	3impia 830	1 |- ((A e. R /\ B e. S /\ D e. T) -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  <.cop 2411

This theorem is referenced by:  ltxrt 5495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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