Theorem eupth2lem2 27079

Description: Lemma for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.)

Hypothesis

Ref	Expression
eupth2lem2.1	|- B e. _V

Assertion

Ref	Expression
eupth2lem2	|- ( ( B =/= C /\ B = U ) -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Proof of Theorem eupth2lem2

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . . . . 7 |- ( ( B =/= C /\ B = U ) -> B = B )
2	1	olcd 408	. . . . . 6 |- ( ( B =/= C /\ B = U ) -> ( B = A \/ B = B ) )
3	2	biantrud 528	. . . . 5 |- ( ( B =/= C /\ B = U ) -> ( A =/= B <-> ( A =/= B /\ ( B = A \/ B = B ) ) ) )
4		eupth2lem2.1	. . . . . 6 |- B e. _V
5		eupth2lem1 27078	. . . . . 6 |- ( B e. _V -> ( B e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( B = A \/ B = B ) ) ) )
6	4, 5	ax-mp 5	. . . . 5 |- ( B e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( B = A \/ B = B ) ) )
7	3, 6	syl6bbr 278	. . . 4 |- ( ( B =/= C /\ B = U ) -> ( A =/= B <-> B e. if ( A = B , (/) , { A , B } ) ) )
8		simpr 477	. . . . 5 |- ( ( B =/= C /\ B = U ) -> B = U )
9	8	eleq1d 2686	. . . 4 |- ( ( B =/= C /\ B = U ) -> ( B e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = B , (/) , { A , B } ) ) )
10	7, 9	bitrd 268	. . 3 |- ( ( B =/= C /\ B = U ) -> ( A =/= B <-> U e. if ( A = B , (/) , { A , B } ) ) )
11	10	necon1bbid 2833	. 2 |- ( ( B =/= C /\ B = U ) -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> A = B ) )
12		simpl 473	. . . . . . 7 |- ( ( B =/= C /\ B = U ) -> B =/= C )
13		neeq1 2856	. . . . . . 7 |- ( B = A -> ( B =/= C <-> A =/= C ) )
14	12, 13	syl5ibcom 235	. . . . . 6 |- ( ( B =/= C /\ B = U ) -> ( B = A -> A =/= C ) )
15	14	pm4.71rd 667	. . . . 5 |- ( ( B =/= C /\ B = U ) -> ( B = A <-> ( A =/= C /\ B = A ) ) )
16		eqcom 2629	. . . . 5 |- ( A = B <-> B = A )
17		ancom 466	. . . . 5 |- ( ( B = A /\ A =/= C ) <-> ( A =/= C /\ B = A ) )
18	15, 16, 17	3bitr4g 303	. . . 4 |- ( ( B =/= C /\ B = U ) -> ( A = B <-> ( B = A /\ A =/= C ) ) )
19	12	neneqd 2799	. . . . . . 7 |- ( ( B =/= C /\ B = U ) -> -. B = C )
20		biorf 420	. . . . . . 7 |- ( -. B = C -> ( B = A <-> ( B = C \/ B = A ) ) )
21	19, 20	syl 17	. . . . . 6 |- ( ( B =/= C /\ B = U ) -> ( B = A <-> ( B = C \/ B = A ) ) )
22		orcom 402	. . . . . 6 |- ( ( B = C \/ B = A ) <-> ( B = A \/ B = C ) )
23	21, 22	syl6bb 276	. . . . 5 |- ( ( B =/= C /\ B = U ) -> ( B = A <-> ( B = A \/ B = C ) ) )
24	23	anbi1d 741	. . . 4 |- ( ( B =/= C /\ B = U ) -> ( ( B = A /\ A =/= C ) <-> ( ( B = A \/ B = C ) /\ A =/= C ) ) )
25	18, 24	bitrd 268	. . 3 |- ( ( B =/= C /\ B = U ) -> ( A = B <-> ( ( B = A \/ B = C ) /\ A =/= C ) ) )
26		ancom 466	. . 3 |- ( ( A =/= C /\ ( B = A \/ B = C ) ) <-> ( ( B = A \/ B = C ) /\ A =/= C ) )
27	25, 26	syl6bbr 278	. 2 |- ( ( B =/= C /\ B = U ) -> ( A = B <-> ( A =/= C /\ ( B = A \/ B = C ) ) ) )
28		eupth2lem1 27078	. . . 4 |- ( B e. _V -> ( B e. if ( A = C , (/) , { A , C } ) <-> ( A =/= C /\ ( B = A \/ B = C ) ) ) )
29	4, 28	ax-mp 5	. . 3 |- ( B e. if ( A = C , (/) , { A , C } ) <-> ( A =/= C /\ ( B = A \/ B = C ) ) )
30	8	eleq1d 2686	. . 3 |- ( ( B =/= C /\ B = U ) -> ( B e. if ( A = C , (/) , { A , C } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )
31	29, 30	syl5bbr 274	. 2 |- ( ( B =/= C /\ B = U ) -> ( ( A =/= C /\ ( B = A \/ B = C ) ) <-> U e. if ( A = C , (/) , { A , C } ) ) )
32	11, 27, 31	3bitrd 294	1 |- ( ( B =/= C /\ B = U ) -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ifcif 4086   {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-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180

This theorem is referenced by:  eupth2lem3lem4  27091