Theorem opthg2 4251

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 4250	. 2 |- ( ( C e. V /\ D e. W ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
2		eqcom 2189	. 2 |- ( <. A , B >. = <. C , D >. <-> <. C , D >. = <. A , B >. )
3		eqcom 2189	. . 3 |- ( A = C <-> C = A )
4		eqcom 2189	. . 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 1363   e. wcel 2158   <.cop 3607

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613

This theorem is referenced by:  opth2  4252  fliftel  5807  axprecex  7893