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Theorem bnj581 29216
Description: Technical lemma for bnj580 29221. 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 3208 . 2  |-  ( [. g  /  f ]. ch  <->  [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )
)
4 vex 2951 . . . 4  |-  g  e. 
_V
5 sbc3ang 3211 . . . 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 29022 . . . . 5  |-  ( [. g  /  f ]. f  Fn  n  <->  g  Fn  n
)
87bicomi 194 . . . 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 1142 . . 3  |-  ( ( g  Fn  n  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) )
126, 11bitr4i 244 . 2  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
131, 3, 123bitri 263 1  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    e. wcel 1725   _Vcvv 2948   [.wsbc 3153    Fn wfn 5441
This theorem is referenced by:  bnj580  29221  bnj849  29233
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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