Theorem opthg 2784

Description: Ordered pair theorem.

Hypotheses

Ref	Expression
opthg.1	|- A e. V
opthg.2	|- B e. V

Assertion

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

Proof of Theorem opthg

Step	Hyp	Ref	Expression
1		opeq2 2485	. . 3 |- (x = D -> <.C, x>. = <.C, D>.)
2	1	eqeq2d 1484	. 2 |- (x = D -> (<.A, B>. = <.C, x>. <-> <.A, B>. = <.C, D>.))
3		eqeq2 1482	. . 3 |- (x = D -> (B = x <-> B = D))
4	3	anbi2d 615	. 2 |- (x = D -> ((A = C /\ B = x) <-> (A = C /\ B = D)))
5		opthg.1	. . 3 |- A e. V
6		opthg.2	. . 3 |- B e. V
7		visset 1810	. . 3 |- x e. V
8	5, 6, 7	opth 2783	. 2 |- (<.A, B>. = <.C, x>. <-> (A = C /\ B = x))
9	2, 4, 8	vtoclbg 1845	1 |- (D e. R -> (<.A, B>. = <.C, D>. <-> (A = C /\ B = D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808  <.cop 2408

This theorem is referenced by:  opthgg 2785  otthg 2786  copsex4g 2790  dmsnop 3324  elo 10403

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413

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