Theorem opthg2 4355

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 4354	. 2 |- ( ( C e. V /\ D e. W ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
2		eqcom 2234	. 2 |- ( <. A , B >. = <. C , D >. <-> <. C , D >. = <. A , B >. )
3		eqcom 2234	. . 3 |- ( A = C <-> C = A )
4		eqcom 2234	. . 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 1398   e. wcel 2203   <.cop 3692

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698

This theorem is referenced by:  opth2  4356  fliftel  5966  axprecex  8195