Theorem opthg2 4241

Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opthg2	|- ( ( C e. V /\ D e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Proof of Theorem opthg2

Step	Hyp	Ref	Expression
1		opthg 4240	. 2 |- ( ( C e. V /\ D e. W ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
2		eqcom 2179	. 2 |- ( <. A , B >. = <. C , D >. <-> <. C , D >. = <. A , B >. )
3		eqcom 2179	. . 3 |- ( A = C <-> C = A )
4		eqcom 2179	. . 3 |- ( B = D <-> D = B )
5	3, 4	anbi12i 460	. 2 |- ( ( A = C /\ B = D ) <-> ( C = A /\ D = B ) )
6	1, 2, 5	3bitr4g 223	1 |- ( ( C e. V /\ D e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   <.cop 3597

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603

This theorem is referenced by:  opth2  4242  fliftel  5796  axprecex  7881