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Theorem bnj1173 28702
Description: Technical lemma for bnj69 28710. 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
bnj1173.3  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
bnj1173.5  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
bnj1173.9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
bnj1173.17  |-  ( (
ph  /\  ps )  ->  X  e.  A )
Assertion
Ref Expression
bnj1173  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  w  e.  A ) )

Proof of Theorem bnj1173
StepHypRef Expression
1 bnj1173.5 . . 3  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
2 3simpc 956 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  ->  ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )
3 bnj1173.9 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
433adant3 977 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  R  FrSe  A )
5 bnj1173.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
653adant3 977 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  X  e.  A )
7 elin 3466 . . . . . . . . 9  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  <->  ( z  e. 
trCl ( X ,  A ,  R )  /\  z  e.  B
) )
87simplbi 447 . . . . . . . 8  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  z  e.  trCl ( X ,  A ,  R ) )
9 bnj1173.3 . . . . . . . 8  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
108, 9eleq2s 2472 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  trCl ( X ,  A ,  R )
)
11103ad2ant3 980 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  trCl ( X ,  A ,  R )
)
12 pm3.21 436 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R )
) ) ) )
134, 6, 11, 12syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R )
) ) ) )
14 bnj170 28393 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  <->  ( ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e. 
trCl ( X ,  A ,  R )
) ) )
1513, 14syl6ibr 219 . . . 4  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) ) )
162, 15impbid2 196 . . 3  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  <->  ( ( R 
FrSe  A  /\  z  e.  A )  /\  w  e.  A ) ) )
171, 16syl5bb 249 . 2  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  ( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A ) ) )
18 bnj1147 28694 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
1918, 11bnj1213 28501 . . . 4  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  A )
204, 19jca 519 . . 3  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( R  FrSe  A  /\  z  e.  A ) )
2120biantrurd 495 . 2  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
w  e.  A  <->  ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) ) )
2217, 21bitr4d 248 1  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  w  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3255    FrSe w-bnj15 28387    trClc-bnj18 28389
This theorem is referenced by:  bnj1190  28708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-iota 5351  df-fn 5390  df-fv 5395  df-bnj17 28382  df-bnj14 28384  df-bnj18 28390
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