Theorem altopthbg 32075

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Assertion

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

Proof of Theorem altopthbg

Step	Hyp	Ref	Expression
1		altopthsn 32068	. 2 |- ( << A , B >> = << C , D >> <-> ( { A } = { C } /\ { B } = { D } ) )
2		sneqbg 4374	. . 3 |- ( A e. V -> ( { A } = { C } <-> A = C ) )
3		sneqbg 4374	. . . 4 |- ( D e. W -> ( { D } = { B } <-> D = B ) )
4		eqcom 2629	. . . 4 |- ( { B } = { D } <-> { D } = { B } )
5		eqcom 2629	. . . 4 |- ( B = D <-> D = B )
6	3, 4, 5	3bitr4g 303	. . 3 |- ( D e. W -> ( { B } = { D } <-> B = D ) )
7	2, 6	bi2anan9 917	. 2 |- ( ( A e. V /\ D e. W ) -> ( ( { A } = { C } /\ { B } = { D } ) <-> ( A = C /\ B = D ) ) )
8	1, 7	syl5bb 272	1 |- ( ( A e. V /\ D e. W ) -> ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <<caltop 32063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-altop 32065

This theorem is referenced by:  altopthb  32077