Theorem prel12 4164

Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preqr1.a	|- A e. _V
preqr1.b	|- B e. _V
preq12b.c	|- C e. _V
preq12b.d	|- D e. _V

Assertion

Ref	Expression
prel12	|- ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) )

Proof of Theorem prel12

Step	Hyp	Ref	Expression
1		preqr1.a	. . . . 5 |- A e. _V
2	1	prid1 4092	. . . 4 |- A e. { A , B }
3		eleq2 2528	. . . 4 |- ( { A , B } = { C , D } -> ( A e. { A , B } <-> A e. { C , D } ) )
4	2, 3	mpbii 216	. . 3 |- ( { A , B } = { C , D } -> A e. { C , D } )
5		preqr1.b	. . . . 5 |- B e. _V
6	5	prid2 4093	. . . 4 |- B e. { A , B }
7		eleq2 2528	. . . 4 |- ( { A , B } = { C , D } -> ( B e. { A , B } <-> B e. { C , D } ) )
8	6, 7	mpbii 216	. . 3 |- ( { A , B } = { C , D } -> B e. { C , D } )
9	4, 8	jca 539	. 2 |- ( { A , B } = { C , D } -> ( A e. { C , D } /\ B e. { C , D } ) )
10	1	elpr 3997	. . . 4 |- ( A e. { C , D } <-> ( A = C \/ A = D ) )
11		eqeq2 2472	. . . . . . . . . . . 12 |- ( B = D -> ( A = B <-> A = D ) )
12	11	notbid 300	. . . . . . . . . . 11 |- ( B = D -> ( -. A = B <-> -. A = D ) )
13		orel2 389	. . . . . . . . . . 11 |- ( -. A = D -> ( ( A = C \/ A = D ) -> A = C ) )
14	12, 13	syl6bi 236	. . . . . . . . . 10 |- ( B = D -> ( -. A = B -> ( ( A = C \/ A = D ) -> A = C ) ) )
15	14	com3l 84	. . . . . . . . 9 |- ( -. A = B -> ( ( A = C \/ A = D ) -> ( B = D -> A = C ) ) )
16	15	imp 435	. . . . . . . 8 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = D -> A = C ) )
17	16	ancrd 561	. . . . . . 7 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = D -> ( A = C /\ B = D ) ) )
18		eqeq2 2472	. . . . . . . . . . . 12 |- ( B = C -> ( A = B <-> A = C ) )
19	18	notbid 300	. . . . . . . . . . 11 |- ( B = C -> ( -. A = B <-> -. A = C ) )
20		orel1 388	. . . . . . . . . . 11 |- ( -. A = C -> ( ( A = C \/ A = D ) -> A = D ) )
21	19, 20	syl6bi 236	. . . . . . . . . 10 |- ( B = C -> ( -. A = B -> ( ( A = C \/ A = D ) -> A = D ) ) )
22	21	com3l 84	. . . . . . . . 9 |- ( -. A = B -> ( ( A = C \/ A = D ) -> ( B = C -> A = D ) ) )
23	22	imp 435	. . . . . . . 8 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = C -> A = D ) )
24	23	ancrd 561	. . . . . . 7 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = C -> ( A = D /\ B = C ) ) )
25	17, 24	orim12d 854	. . . . . 6 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( ( B = D \/ B = C ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	5	elpr 3997	. . . . . . 7 |- ( B e. { C , D } <-> ( B = C \/ B = D ) )
27		orcom 393	. . . . . . 7 |- ( ( B = C \/ B = D ) <-> ( B = D \/ B = C ) )
28	26, 27	bitri 257	. . . . . 6 |- ( B e. { C , D } <-> ( B = D \/ B = C ) )
29		preq12b.c	. . . . . . 7 |- C e. _V
30		preq12b.d	. . . . . . 7 |- D e. _V
31	1, 5, 29, 30	preq12b 4163	. . . . . 6 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
32	25, 28, 31	3imtr4g 278	. . . . 5 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B e. { C , D } -> { A , B } = { C , D } ) )
33	32	ex 440	. . . 4 |- ( -. A = B -> ( ( A = C \/ A = D ) -> ( B e. { C , D } -> { A , B } = { C , D } ) ) )
34	10, 33	syl5bi 225	. . 3 |- ( -. A = B -> ( A e. { C , D } -> ( B e. { C , D } -> { A , B } = { C , D } ) ) )
35	34	impd 437	. 2 |- ( -. A = B -> ( ( A e. { C , D } /\ B e. { C , D } ) -> { A , B } = { C , D } ) )
36	9, 35	impbid2 209	1 |- ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1454   e. wcel 1897   _Vcvv 3056   {cpr 3981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-un 3420  df-sn 3980  df-pr 3982

This theorem is referenced by:  prel12g  4168  dfac2  8586