Theorem opthg2 4156

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 4155	. 2 |- ( ( C e. V /\ D e. W ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
2		eqcom 2139	. 2 |- ( <. A , B >. = <. C , D >. <-> <. C , D >. = <. A , B >. )
3		eqcom 2139	. . 3 |- ( A = C <-> C = A )
4		eqcom 2139	. . 3 |- ( B = D <-> D = B )
5	3, 4	anbi12i 455	. 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 1331   e. wcel 1480   <.cop 3525

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531

This theorem is referenced by:  opth2  4157  fliftel  5687  axprecex  7681