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Theorem bnj1040 29341
Description: Technical lemma for bnj69 29379. 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
bnj1040.1  |-  ( ph'  <->  [. j  /  i ]. ph )
bnj1040.2  |-  ( ps'  <->  [. j  /  i ]. ps )
bnj1040.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1040.4  |-  ( ch'  <->  [. j  /  i ]. ch )
Assertion
Ref Expression
bnj1040  |-  ( ch'  <->  (
n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' ) )
Distinct variable groups:    D, i    f, i    i, n
Allowed substitution hints:    ph( f, i, j, n)    ps( f,
i, j, n)    ch( f, i, j, n)    D( f, j, n)    ph'( f, i, j, n)    ps'( f, i, j, n)    ch'( f, i, j, n)

Proof of Theorem bnj1040
StepHypRef Expression
1 bnj1040.4 . 2  |-  ( ch'  <->  [. j  /  i ]. ch )
2 bnj1040.3 . . 3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
32sbcbii 3216 . 2  |-  ( [. j  /  i ]. ch  <->  [. j  /  i ]. ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
)
4 df-bnj17 29051 . . 3  |-  ( (
[. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph  /\  [. j  /  i ]. ps ) 
<->  ( ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph )  /\  [. j  /  i ]. ps ) )
5 vex 2959 . . . . . 6  |-  j  e. 
_V
65bnj525 29106 . . . . 5  |-  ( [. j  /  i ]. n  e.  D  <->  n  e.  D
)
76bicomi 194 . . . 4  |-  ( n  e.  D  <->  [. j  / 
i ]. n  e.  D
)
85bnj525 29106 . . . . 5  |-  ( [. j  /  i ]. f  Fn  n  <->  f  Fn  n
)
98bicomi 194 . . . 4  |-  ( f  Fn  n  <->  [. j  / 
i ]. f  Fn  n
)
10 bnj1040.1 . . . 4  |-  ( ph'  <->  [. j  /  i ]. ph )
11 bnj1040.2 . . . 4  |-  ( ps'  <->  [. j  /  i ]. ps )
127, 9, 10, 11bnj887 29134 . . 3  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' )  <->  ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph  /\  [. j  /  i ]. ps ) )
13 df-bnj17 29051 . . . . 5  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( (
n  e.  D  /\  f  Fn  n  /\  ph )  /\  ps )
)
1413sbcbii 3216 . . . 4  |-  ( [. j  /  i ]. (
n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  [. j  / 
i ]. ( ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  ps ) )
15 sbcan 3203 . . . 4  |-  ( [. j  /  i ]. (
( n  e.  D  /\  f  Fn  n  /\  ph )  /\  ps ) 
<->  ( [. j  / 
i ]. ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  [. j  /  i ]. ps ) )
16 sbc3ang 3219 . . . . . 6  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ( n  e.  D  /\  f  Fn  n  /\  ph )  <->  ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph )
) )
175, 16ax-mp 8 . . . . 5  |-  ( [. j  /  i ]. (
n  e.  D  /\  f  Fn  n  /\  ph )  <->  ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph )
)
1817anbi1i 677 . . . 4  |-  ( (
[. j  /  i ]. ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  [. j  /  i ]. ps ) 
<->  ( ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph )  /\  [. j  /  i ]. ps ) )
1914, 15, 183bitri 263 . . 3  |-  ( [. j  /  i ]. (
n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( ( [. j  /  i ]. n  e.  D  /\  [. j  /  i ]. f  Fn  n  /\  [. j  /  i ]. ph )  /\  [. j  /  i ]. ps ) )
204, 12, 193bitr4ri 270 . 2  |-  ( [. j  /  i ]. (
n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
211, 3, 203bitri 263 1  |-  ( ch'  <->  (
n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   _Vcvv 2956   [.wsbc 3161    Fn wfn 5449    /\ w-bnj17 29050
This theorem is referenced by:  bnj1128  29359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-bnj17 29051
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