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Theorem bnj581 29256
Description: Technical lemma for bnj580 29261. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj581.3  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj581.4  |-  ( ph'  <->  [. g  /  f ]. ph )
bnj581.5  |-  ( ps'  <->  [. g  /  f ]. ps )
bnj581.6  |-  ( ch'  <->  [. g  /  f ]. ch )
Assertion
Ref Expression
bnj581  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
Distinct variable group:    f, n
Allowed substitution hints:    ph( f, g, n)    ps( f, g, n)    ch( f, g, n)    ph'( f, g, n)    ps'( f, g, n)    ch'( f, g, n)

Proof of Theorem bnj581
StepHypRef Expression
1 bnj581.6 . 2  |-  ( ch'  <->  [. g  /  f ]. ch )
2 bnj581.3 . . 3  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
32sbcbii 3059 . 2  |-  ( [. g  /  f ]. ch  <->  [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )
)
4 vex 2804 . . . 4  |-  g  e. 
_V
5 sbc3ang 3062 . . . 4  |-  ( g  e.  _V  ->  ( [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )  <->  (
[. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) ) )
64, 5ax-mp 8 . . 3  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) )
7 bnj62 29062 . . . . 5  |-  ( [. g  /  f ]. f  Fn  n  <->  g  Fn  n
)
87bicomi 193 . . . 4  |-  ( g  Fn  n  <->  [. g  / 
f ]. f  Fn  n
)
9 bnj581.4 . . . 4  |-  ( ph'  <->  [. g  /  f ]. ph )
10 bnj581.5 . . . 4  |-  ( ps'  <->  [. g  /  f ]. ps )
118, 9, 103anbi123i 1140 . . 3  |-  ( ( g  Fn  n  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) )
126, 11bitr4i 243 . 2  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
131, 3, 123bitri 262 1  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    e. wcel 1696   _Vcvv 2801   [.wsbc 3004    Fn wfn 5266
This theorem is referenced by:  bnj580  29261  bnj849  29273
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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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