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Theorem bnj1172 33764
Description: Technical lemma for bnj69 33773. 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 317 . . . . . . 7  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( th  ->  (
w R z  ->  -.  w  e.  B
) )  <->  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B )
) ) )
43pm5.32i 637 . . . . . 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 312 . . . . 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 1625 . . . 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 1652 . . 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 208 . 2  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
( ph  /\  ps  /\  z  e.  C )  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
9 simp3 997 . . . . . . 7  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  C )
10 bnj1172.3 . . . . . . 7  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
119, 10syl6eleq 2539 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  (  trCl ( X ,  A ,  R
)  i^i  B )
)
12 elin 3669 . . . . . . 7  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  <->  ( z  e. 
trCl ( X ,  A ,  R )  /\  z  e.  B
) )
1312simprbi 464 . . . . . 6  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  z  e.  B )
1411, 13syl 16 . . . . 5  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  B )
1514anim1i 568 . . . 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 14 . . 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 1618 . 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 33484 1  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972   A.wal 1379    = wceq 1381   E.wex 1597    e. wcel 1802    i^i cin 3457   class class class wbr 4433    trClc-bnj18 33454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-in 3465
This theorem is referenced by:  bnj1190  33771
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