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Metamath Proof Explorer

Theorem bnj1172

Description: Technical lemma for bnj69 . 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.)

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

Proof

Step Hyp Ref Expression
1 bnj1172.3 
 |-  C = ( _trCl ( X , A , R ) i^i B )
2 bnj1172.96 
 |-  E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( th -> ( w R z -> -. w e. B ) ) ) )
3 bnj1172.113 
 |-  ( ( ph /\ ps /\ z e. C ) -> ( th <-> w e. A ) )
4 3 imbi1d 
 |-  ( ( ph /\ ps /\ z e. C ) -> ( ( th -> ( w R z -> -. w e. B ) ) <-> ( w e. A -> ( w R z -> -. w e. B ) ) ) )
5 4 pm5.32i 
 |-  ( ( ( 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 ) ) ) )
6 5 imbi2i 
 |-  ( ( ( 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 ) ) ) ) )
7 6 albii 
 |-  ( 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 ) ) ) ) )
8 7 exbii 
 |-  ( 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 ) ) ) ) )
9 2 8 mpbi 
 |-  E. z A. w ( ( ph /\ ps ) -> ( ( ph /\ ps /\ z e. C ) /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )
10 simp3 
 |-  ( ( ph /\ ps /\ z e. C ) -> z e. C )
11 10 1 eleqtrdi 
 |-  ( ( ph /\ ps /\ z e. C ) -> z e. ( _trCl ( X , A , R ) i^i B ) )
12 11 elin2d 
 |-  ( ( ph /\ ps /\ z e. C ) -> z e. B )
13 12 anim1i 
 |-  ( ( ( 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 
 |-  ( ( ( 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 
 |-  ( 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 9 15 bnj101 
 |-  E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )