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Theorem bnj1172 29347
Description: Technical lemma for bnj69 29356. 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
StepHypRef 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 ) )
32imbi1d 308 . . . . . . 7  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( th  ->  (
w R z  ->  -.  w  e.  B
) )  <->  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B )
) ) )
43pm5.32i 618 . . . . . 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 ) ) ) )
54imbi2i 303 . . . . 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 ) ) ) ) )
65albii 1556 . . . 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 ) ) ) ) )
76exbii 1572 . . 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 ) ) ) ) )
81, 7mpbi 199 . 2  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
( ph  /\  ps  /\  z  e.  C )  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
9 simp3 957 . . . . . . 7  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  C )
10 bnj1172.3 . . . . . . 7  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
119, 10syl6eleq 2386 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  (  trCl ( X ,  A ,  R
)  i^i  B )
)
12 elin 3371 . . . . . . 7  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  <->  ( z  e. 
trCl ( X ,  A ,  R )  /\  z  e.  B
) )
1312simprbi 450 . . . . . 6  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  z  e.  B )
1411, 13syl 15 . . . . 5  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  B )
1514anim1i 551 . . . 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 ) ) ) )
1615imim2i 13 . . 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 ) ) ) ) )
1716alimi 1549 . 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 ) ) ) ) )
188, 17bnj101 29065 1  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696    i^i cin 3164   class class class wbr 4039    trClc-bnj18 29035
This theorem is referenced by:  bnj1190  29354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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