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Theorem bnj1171 33153
Description: Technical lemma for bnj69 33163. 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
bnj1171.13  |-  ( (
ph  /\  ps )  ->  B  C_  A )
bnj1171.129  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
Assertion
Ref Expression
bnj1171  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )

Proof of Theorem bnj1171
StepHypRef Expression
1 bnj1171.129 . 2  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
2 bnj1171.13 . . . . . . . . . . 11  |-  ( (
ph  /\  ps )  ->  B  C_  A )
32sseld 3503 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  ( w  e.  B  ->  w  e.  A ) )
43pm4.71rd 635 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  ( w  e.  B  <->  ( w  e.  A  /\  w  e.  B )
) )
54imbi1d 317 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ( ( w  e.  B  ->  -.  w R z )  <->  ( (
w  e.  A  /\  w  e.  B )  ->  -.  w R z ) ) )
6 impexp 446 . . . . . . . 8  |-  ( ( ( w  e.  A  /\  w  e.  B
)  ->  -.  w R z )  <->  ( w  e.  A  ->  ( w  e.  B  ->  -.  w R z ) ) )
75, 6syl6bb 261 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ( w  e.  B  ->  -.  w R z )  <->  ( w  e.  A  ->  ( w  e.  B  ->  -.  w R z ) ) ) )
8 con2b 334 . . . . . . . 8  |-  ( ( w R z  ->  -.  w  e.  B
)  <->  ( w  e.  B  ->  -.  w R z ) )
98imbi2i 312 . . . . . . 7  |-  ( ( w  e.  A  -> 
( w R z  ->  -.  w  e.  B ) )  <->  ( w  e.  A  ->  ( w  e.  B  ->  -.  w R z ) ) )
107, 9syl6bbr 263 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( ( w  e.  B  ->  -.  w R z )  <->  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B )
) ) )
1110anbi2d 703 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) )  <-> 
( z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) ) )
1211pm5.74i 245 . . . 4  |-  ( ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )  <->  ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) ) )
1312albii 1620 . . 3  |-  ( A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )  <->  A. w
( ( ph  /\  ps )  ->  ( z  e.  B  /\  (
w  e.  A  -> 
( w R z  ->  -.  w  e.  B ) ) ) ) )
1413exbii 1644 . 2  |-  ( E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )  <->  E. z A. w
( ( ph  /\  ps )  ->  ( z  e.  B  /\  (
w  e.  A  -> 
( w R z  ->  -.  w  e.  B ) ) ) ) )
151, 14mpbir 209 1  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596    e. wcel 1767    C_ wss 3476   class class class wbr 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490
This theorem is referenced by:  bnj1190  33161
  Copyright terms: Public domain W3C validator