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Theorem bnj581 28940
Description: Technical lemma for bnj580 28945. 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 3046 . 2  |-  ( [. g  /  f ]. ch  <->  [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )
)
4 vex 2791 . . . 4  |-  g  e. 
_V
5 sbc3ang 3049 . . . 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 28746 . . . . 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 1684   _Vcvv 2788   [.wsbc 2991    Fn wfn 5250
This theorem is referenced by:  bnj580  28945  bnj849  28957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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