Theorem bnj1172 31069

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1172.3	|- C = ( trCl ( X , A , R ) i^i B )
bnj1172.96	|- E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) )
bnj1172.113	|- ( ( ph /\ ps /\ z e. C ) -> ( th <-> w e. A ) )

Assertion

Ref	Expression
bnj1172	|- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )

Proof of Theorem bnj1172

Step	Hyp	Ref	Expression
1		bnj1172.96	. . 3 |- E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) )
2		bnj1172.113	. . . . . . . 8 |- ( ( ph /\ ps /\ z e. C ) -> ( th <-> w e. A ) )
3	2	imbi1d 331	. . . . . . 7 |- ( ( ph /\ ps /\ z e. C ) -> ( ( th -> ( w R z -> -. w e. B ) ) <-> ( w e. A -> ( w R z -> -. w e. B ) ) ) )
4	3	pm5.32i 669	. . . . . 6 |- ( ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) <-> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )
5	4	imbi2i 326	. . . . 5 |- ( ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) ) <-> ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
6	5	albii 1747	. . . 4 |- ( A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) ) <-> A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
7	6	exbii 1774	. . 3 |- ( E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) ) <-> E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
8	1, 7	mpbi 220	. 2 |- E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )
9		simp3 1063	. . . . . . 7 |- ( ( ph /\ ps /\ z e. C ) -> z e. C )
10		bnj1172.3	. . . . . . 7 |- C = ( trCl ( X , A , R ) i^i B )
11	9, 10	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ ps /\ z e. C ) -> z e. ( trCl ( X , A , R ) i^i B ) )
12	11	elin2d 3803	. . . . 5 |- ( ( ph /\ ps /\ z e. C ) -> z e. B )
13	12	anim1i 592	. . . 4 |- ( ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )
14	13	imim2i 16	. . 3 |- ( ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) -> ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
15	14	alimi 1739	. 2 |- ( A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) -> A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
16	8, 15	bnj101 30789	1 |- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   i^i cin 3573   class class class wbr 4653   trClc-bnj18 30760

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-3an 1039  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-in 3581

This theorem is referenced by:  bnj1190  31076