Theorem opthg 4268

Description: Ordered pair theorem. C and D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem opthg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3805	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	eqeq1d 2202	. . 3 |- ( x = A -> ( <. x , y >. = <. C , D >. <-> <. A , y >. = <. C , D >. ) )
3		eqeq1 2200	. . . 4 |- ( x = A -> ( x = C <-> A = C ) )
4	3	anbi1d 465	. . 3 |- ( x = A -> ( ( x = C /\ y = D ) <-> ( A = C /\ y = D ) ) )
5	2, 4	bibi12d 235	. 2 |- ( x = A -> ( ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) ) <-> ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) ) )
6		opeq2 3806	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
7	6	eqeq1d 2202	. . 3 |- ( y = B -> ( <. A , y >. = <. C , D >. <-> <. A , B >. = <. C , D >. ) )
8		eqeq1 2200	. . . 4 |- ( y = B -> ( y = D <-> B = D ) )
9	8	anbi2d 464	. . 3 |- ( y = B -> ( ( A = C /\ y = D ) <-> ( A = C /\ B = D ) ) )
10	7, 9	bibi12d 235	. 2 |- ( y = B -> ( ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) <-> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) )
11		vex 2763	. . 3 |- x e. _V
12		vex 2763	. . 3 |- y e. _V
13	11, 12	opth 4267	. 2 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
14	5, 10, 13	vtocl2g 2825	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   <.cop 3622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628

This theorem is referenced by:  opthg2  4269  xpopth  6231  eqop  6232  inl11  7126  preqlu  7534  cauappcvgprlemladd  7720  elrealeu  7891  qnumdenbi  12333  crth  12365  imasaddfnlemg  12900