Theorem prel12 4383

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 4297	. . . 4 |- A e. { A , B }
3		eleq2 2690	. . . 4 |- ( { A , B } = { C , D } -> ( A e. { A , B } <-> A e. { C , D } ) )
4	2, 3	mpbii 223	. . 3 |- ( { A , B } = { C , D } -> A e. { C , D } )
5		preqr1.b	. . . . 5 |- B e. _V
6	5	prid2 4298	. . . 4 |- B e. { A , B }
7		eleq2 2690	. . . 4 |- ( { A , B } = { C , D } -> ( B e. { A , B } <-> B e. { C , D } ) )
8	6, 7	mpbii 223	. . 3 |- ( { A , B } = { C , D } -> B e. { C , D } )
9	4, 8	jca 554	. 2 |- ( { A , B } = { C , D } -> ( A e. { C , D } /\ B e. { C , D } ) )
10	1	elpr 4198	. . . 4 |- ( A e. { C , D } <-> ( A = C \/ A = D ) )
11		eqeq2 2633	. . . . . . . . . . . 12 |- ( B = D -> ( A = B <-> A = D ) )
12	11	notbid 308	. . . . . . . . . . 11 |- ( B = D -> ( -. A = B <-> -. A = D ) )
13		orel2 398	. . . . . . . . . . 11 |- ( -. A = D -> ( ( A = C \/ A = D ) -> A = C ) )
14	12, 13	syl6bi 243	. . . . . . . . . 10 |- ( B = D -> ( -. A = B -> ( ( A = C \/ A = D ) -> A = C ) ) )
15	14	impd 447	. . . . . . . . 9 |- ( B = D -> ( ( -. A = B /\ ( A = C \/ A = D ) ) -> A = C ) )
16	15	com12 32	. . . . . . . 8 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = D -> A = C ) )
17	16	ancrd 577	. . . . . . 7 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = D -> ( A = C /\ B = D ) ) )
18		eqeq2 2633	. . . . . . . . . . . 12 |- ( B = C -> ( A = B <-> A = C ) )
19	18	notbid 308	. . . . . . . . . . 11 |- ( B = C -> ( -. A = B <-> -. A = C ) )
20		orel1 397	. . . . . . . . . . 11 |- ( -. A = C -> ( ( A = C \/ A = D ) -> A = D ) )
21	19, 20	syl6bi 243	. . . . . . . . . 10 |- ( B = C -> ( -. A = B -> ( ( A = C \/ A = D ) -> A = D ) ) )
22	21	impd 447	. . . . . . . . 9 |- ( B = C -> ( ( -. A = B /\ ( A = C \/ A = D ) ) -> A = D ) )
23	22	com12 32	. . . . . . . 8 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = C -> A = D ) )
24	23	ancrd 577	. . . . . . 7 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B = C -> ( A = D /\ B = C ) ) )
25	17, 24	orim12d 883	. . . . . 6 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( ( B = D \/ B = C ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	5	elpr 4198	. . . . . . 7 |- ( B e. { C , D } <-> ( B = C \/ B = D ) )
27		orcom 402	. . . . . . 7 |- ( ( B = C \/ B = D ) <-> ( B = D \/ B = C ) )
28	26, 27	bitri 264	. . . . . 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 4382	. . . . . 6 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
32	25, 28, 31	3imtr4g 285	. . . . 5 |- ( ( -. A = B /\ ( A = C \/ A = D ) ) -> ( B e. { C , D } -> { A , B } = { C , D } ) )
33	32	ex 450	. . . 4 |- ( -. A = B -> ( ( A = C \/ A = D ) -> ( B e. { C , D } -> { A , B } = { C , D } ) ) )
34	10, 33	syl5bi 232	. . 3 |- ( -. A = B -> ( A e. { C , D } -> ( B e. { C , D } -> { A , B } = { C , D } ) ) )
35	34	impd 447	. 2 |- ( -. A = B -> ( ( A e. { C , D } /\ B e. { C , D } ) -> { A , B } = { C , D } ) )
36	9, 35	impbid2 216	1 |- ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179

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

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-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  prel12g  4387  dfac2  8953