Theorem prel12 3533

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

Hypotheses

Ref	Expression
preq12b.1	|- A e. _V
preq12b.2	|- B e. _V
preq12b.3	|- C e. _V
preq12b.4	|- 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		preq12b.1	. . . . 5 |- A e. _V
2	1	prid1 3467	. . . 4 |- A e. {A , B }
3		eleq2 2098	. . . 4 |- ( {A , B } = { C , D } -> (A e. {A , B } <-> A e. { C , D }))
4	2, 3	mpbii 136	. . 3 |- ( {A , B } = { C , D } -> A e. { C , D })
5		preq12b.2	. . . . 5 |- B e. _V
6	5	prid2 3468	. . . 4 |- B e. {A , B }
7		eleq2 2098	. . . 4 |- ( {A , B } = { C , D } -> (B e. {A , B } <-> B e. { C , D }))
8	6, 7	mpbii 136	. . 3 |- ( {A , B } = { C , D } -> B e. { C , D })
9	4, 8	jca 290	. 2 |- ( {A , B } = { C , D } -> (A e. { C , D } /\ B e. { C , D }))
10	1	elpr 3385	. . . 4 |- (A e. { C , D } <-> (A = C \/ A = D))
11		eqeq2 2046	. . . . . . . . . . . 12 |- (B = D -> (A = B <-> A = D))
12	11	notbid 591	. . . . . . . . . . 11 |- (B = D -> (-. A = B <-> -. A = D))
13		orel2 644	. . . . . . . . . . 11 |- (-. A = D -> ((A = C \/ A = D) -> A = C))
14	12, 13	syl6bi 152	. . . . . . . . . 10 |- (B = D -> (-. A = B -> ((A = C \/ A = D) -> A = C)))
15	14	com3l 75	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = D -> A = C)))
16	15	imp 115	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> A = C))
17	16	ancrd 309	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> (A = C /\ B = D)))
18		eqeq2 2046	. . . . . . . . . . . 12 |- (B = C -> (A = B <-> A = C))
19	18	notbid 591	. . . . . . . . . . 11 |- (B = C -> (-. A = B <-> -. A = C))
20		orel1 643	. . . . . . . . . . 11 |- (-. A = C -> ((A = C \/ A = D) -> A = D))
21	19, 20	syl6bi 152	. . . . . . . . . 10 |- (B = C -> (-. A = B -> ((A = C \/ A = D) -> A = D)))
22	21	com3l 75	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = C -> A = D)))
23	22	imp 115	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> A = D))
24	23	ancrd 309	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> (A = D /\ B = C)))
25	17, 24	orim12d 699	. . . . . 6 |- ((-. A = B /\ (A = C \/ A = D)) -> ((B = D \/ B = C) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
26	5	elpr 3385	. . . . . . 7 |- (B e. { C , D } <-> (B = C \/ B = D))
27		orcom 646	. . . . . . 7 |- ((B = C \/ B = D) <-> (B = D \/ B = C))
28	26, 27	bitri 173	. . . . . 6 |- (B e. { C , D } <-> (B = D \/ B = C))
29		preq12b.3	. . . . . . 7 |- C e. _V
30		preq12b.4	. . . . . . 7 |- D e. _V
31	1, 5, 29, 30	preq12b 3532	. . . . . 6 |- ( {A , B } = { C , D } <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))
32	25, 28, 31	3imtr4g 194	. . . . 5 |- ((-. A = B /\ (A = C \/ A = D)) -> (B e. { C , D } -> {A , B } = { C , D }))
33	32	ex 108	. . . 4 |- (-. A = B -> ((A = C \/ A = D) -> (B e. { C , D } -> {A , B } = { C , D })))
34	10, 33	syl5bi 141	. . 3 |- (-. A = B -> (A e. { C , D } -> (B e. { C , D } -> {A , B } = { C , D })))
35	34	impd 242	. 2 |- (-. A = B -> ((A e. { C , D } /\ B e. { C , D }) -> {A , B } = { C , D }))
36	9, 35	impbid2 131	1 |- (-. A = B -> ( {A , B } = { C , D } <-> (A e. { C , D } /\ B e. { C , D })))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628   = wceq 1242   e. wcel 1390   _Vcvv 2551   {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by: (None)