Theorem eupth2lem2dc 16383

Description: Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

Hypotheses

Ref	Expression
eupth2lem2dc.1	|- ( ph -> B e. X )
eupth2lem2dc.dc	|- ( ph -> DECID A = B )
eupth2lem2dc.bc	|- ( ph -> B =/= C )
eupth2lem2dc.bu	|- ( ph -> B = U )

Assertion

Ref	Expression
eupth2lem2dc	|- ( ph -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Proof of Theorem eupth2lem2dc

Step	Hyp	Ref	Expression
1		eupth2lem2dc.dc	. . 3 |- ( ph -> DECID A = B )
2		eqidd 2232	. . . . . . . 8 |- ( ph -> B = B )
3	2	olcd 742	. . . . . . 7 |- ( ph -> ( B = A \/ B = B ) )
4	3	biantrud 304	. . . . . 6 |- ( ph -> ( A =/= B <-> ( A =/= B /\ ( B = A \/ B = B ) ) ) )
5		eupth2lem2dc.1	. . . . . . 7 |- ( ph -> B e. X )
6		eupth2lem1 16382	. . . . . . 7 |- ( B e. X -> ( B e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( B = A \/ B = B ) ) ) )
7	5, 6	syl 14	. . . . . 6 |- ( ph -> ( B e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( B = A \/ B = B ) ) ) )
8		eupth2lem2dc.bu	. . . . . . 7 |- ( ph -> B = U )
9	8	eleq1d 2300	. . . . . 6 |- ( ph -> ( B e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = B , (/) , { A , B } ) ) )
10	4, 7, 9	3bitr2d 216	. . . . 5 |- ( ph -> ( A =/= B <-> U e. if ( A = B , (/) , { A , B } ) ) )
11	10	a1d 22	. . . 4 |- ( ph -> (DECID A = B -> ( A =/= B <-> U e. if ( A = B , (/) , { A , B } ) ) ) )
12	11	necon1bbiddc 2466	. . 3 |- ( ph -> (DECID A = B -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> A = B ) ) )
13	1, 12	mpd 13	. 2 |- ( ph -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> A = B ) )
14		eupth2lem2dc.bc	. . . . . . 7 |- ( ph -> B =/= C )
15		neeq1 2416	. . . . . . 7 |- ( B = A -> ( B =/= C <-> A =/= C ) )
16	14, 15	syl5ibcom 155	. . . . . 6 |- ( ph -> ( B = A -> A =/= C ) )
17	16	pm4.71rd 394	. . . . 5 |- ( ph -> ( B = A <-> ( A =/= C /\ B = A ) ) )
18		eqcom 2233	. . . . 5 |- ( A = B <-> B = A )
19		ancom 266	. . . . 5 |- ( ( B = A /\ A =/= C ) <-> ( A =/= C /\ B = A ) )
20	17, 18, 19	3bitr4g 223	. . . 4 |- ( ph -> ( A = B <-> ( B = A /\ A =/= C ) ) )
21	14	neneqd 2424	. . . . . . 7 |- ( ph -> -. B = C )
22		biorf 752	. . . . . . 7 |- ( -. B = C -> ( B = A <-> ( B = C \/ B = A ) ) )
23	21, 22	syl 14	. . . . . 6 |- ( ph -> ( B = A <-> ( B = C \/ B = A ) ) )
24		orcom 736	. . . . . 6 |- ( ( B = C \/ B = A ) <-> ( B = A \/ B = C ) )
25	23, 24	bitrdi 196	. . . . 5 |- ( ph -> ( B = A <-> ( B = A \/ B = C ) ) )
26	25	anbi1d 465	. . . 4 |- ( ph -> ( ( B = A /\ A =/= C ) <-> ( ( B = A \/ B = C ) /\ A =/= C ) ) )
27	20, 26	bitrd 188	. . 3 |- ( ph -> ( A = B <-> ( ( B = A \/ B = C ) /\ A =/= C ) ) )
28	27	biancomd 271	. 2 |- ( ph -> ( A = B <-> ( A =/= C /\ ( B = A \/ B = C ) ) ) )
29		eupth2lem1 16382	. . . 4 |- ( B e. X -> ( B e. if ( A = C , (/) , { A , C } ) <-> ( A =/= C /\ ( B = A \/ B = C ) ) ) )
30	5, 29	syl 14	. . 3 |- ( ph -> ( B e. if ( A = C , (/) , { A , C } ) <-> ( A =/= C /\ ( B = A \/ B = C ) ) ) )
31	8	eleq1d 2300	. . 3 |- ( ph -> ( B e. if ( A = C , (/) , { A , C } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )
32	30, 31	bitr3d 190	. 2 |- ( ph -> ( ( A =/= C /\ ( B = A \/ B = C ) ) <-> U e. if ( A = C , (/) , { A , C } ) ) )
33	13, 28, 32	3bitrd 214	1 |- ( ph -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   (/)c0 3496   ifcif 3607   {cpr 3674

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-in1 619  ax-in2 620  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-ext 2213

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-if 3608  df-sn 3679  df-pr 3680

This theorem is referenced by:  eupth2lem3lem4fi  16397