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Theorem bnj581 29293
Description: Technical lemma for bnj580 29298. 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 3333 . 2  |-  ( [. g  /  f ]. ch  <->  [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )
)
4 sbc3an 3335 . . 3  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) )
5 bnj62 29100 . . . . 5  |-  ( [. g  /  f ]. f  Fn  n  <->  g  Fn  n
)
65bicomi 202 . . . 4  |-  ( g  Fn  n  <->  [. g  / 
f ]. f  Fn  n
)
7 bnj581.4 . . . 4  |-  ( ph'  <->  [. g  /  f ]. ph )
8 bnj581.5 . . . 4  |-  ( ps'  <->  [. g  /  f ]. ps )
96, 7, 83anbi123i 1186 . . 3  |-  ( ( g  Fn  n  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  n  /\  [. g  /  f ]. ph  /\  [. g  /  f ]. ps ) )
104, 9bitr4i 252 . 2  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
111, 3, 103bitri 271 1  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 974   [.wsbc 3277    Fn wfn 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-fun 5571  df-fn 5572
This theorem is referenced by:  bnj580  29298  bnj849  29310
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