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Theorem bnj873 33710
Description: Technical lemma for bnj69 33794. 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
bnj873.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj873.7  |-  ( ph'  <->  [. g  /  f ]. ph )
bnj873.8  |-  ( ps'  <->  [. g  /  f ]. ps )
Assertion
Ref Expression
bnj873  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Distinct variable groups:    D, f,
g    f, n, g    ph, g    ps, g
Allowed substitution hints:    ph( f, n)    ps( f, n)    B( f,
g, n)    D( n)    ph'( f, g, n)    ps'( f, g, n)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
2 nfv 1694 . . 3  |-  F/ g E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
3 nfcv 2605 . . . 4  |-  F/_ f D
4 nfv 1694 . . . . 5  |-  F/ f  g  Fn  n
5 bnj873.7 . . . . . 6  |-  ( ph'  <->  [. g  /  f ]. ph )
6 nfsbc1v 3333 . . . . . 6  |-  F/ f
[. g  /  f ]. ph
75, 6nfxfr 1632 . . . . 5  |-  F/ f ph'
8 bnj873.8 . . . . . 6  |-  ( ps'  <->  [. g  /  f ]. ps )
9 nfsbc1v 3333 . . . . . 6  |-  F/ f
[. g  /  f ]. ps
108, 9nfxfr 1632 . . . . 5  |-  F/ f ps'
114, 7, 10nf3an 1916 . . . 4  |-  F/ f ( g  Fn  n  /\  ph'  /\  ps' )
123, 11nfrex 2906 . . 3  |-  F/ f E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' )
13 fneq1 5659 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  n  <->  g  Fn  n ) )
14 sbceq1a 3324 . . . . . 6  |-  ( f  =  g  ->  ( ph 
<-> 
[. g  /  f ]. ph ) )
1514, 5syl6bbr 263 . . . . 5  |-  ( f  =  g  ->  ( ph 
<->  ph' ) )
16 sbceq1a 3324 . . . . . 6  |-  ( f  =  g  ->  ( ps 
<-> 
[. g  /  f ]. ps ) )
1716, 8syl6bbr 263 . . . . 5  |-  ( f  =  g  ->  ( ps 
<->  ps' ) )
1813, 15, 173anbi123d 1300 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
1918rexbidv 2954 . . 3  |-  ( f  =  g  ->  ( E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
202, 12, 19cbvab 2584 . 2  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
211, 20eqtri 2472 1  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 974    = wceq 1383   {cab 2428   E.wrex 2794   [.wsbc 3313    Fn wfn 5573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-fun 5580  df-fn 5581
This theorem is referenced by:  bnj849  33711  bnj893  33714
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