Theorem opthg2 4169

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 4168	. 2 |- ( ( C e. V /\ D e. W ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
2		eqcom 2142	. 2 |- ( <. A , B >. = <. C , D >. <-> <. C , D >. = <. A , B >. )
3		eqcom 2142	. . 3 |- ( A = C <-> C = A )
4		eqcom 2142	. . 3 |- ( B = D <-> D = B )
5	3, 4	anbi12i 456	. 2 |- ( ( A = C /\ B = D ) <-> ( C = A /\ D = B ) )
6	1, 2, 5	3bitr4g 222	1 |- ( ( C e. V /\ D e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   <.cop 3535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541

This theorem is referenced by:  opth2  4170  fliftel  5702  axprecex  7712