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Theorem bnj873 29357
Description: Technical lemma for bnj69 29441. 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 1630 . . 3  |-  F/ g E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
3 nfcv 2574 . . . 4  |-  F/_ f D
4 nfv 1630 . . . . 5  |-  F/ f  g  Fn  n
5 bnj873.7 . . . . . 6  |-  ( ph'  <->  [. g  /  f ]. ph )
6 nfsbc1v 3182 . . . . . 6  |-  F/ f
[. g  /  f ]. ph
75, 6nfxfr 1580 . . . . 5  |-  F/ f ph'
8 bnj873.8 . . . . . 6  |-  ( ps'  <->  [. g  /  f ]. ps )
9 nfsbc1v 3182 . . . . . 6  |-  F/ f
[. g  /  f ]. ps
108, 9nfxfr 1580 . . . . 5  |-  F/ f ps'
114, 7, 10nf3an 1850 . . . 4  |-  F/ f ( g  Fn  n  /\  ph'  /\  ps' )
123, 11nfrex 2763 . . 3  |-  F/ f E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' )
13 fneq1 5536 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  n  <->  g  Fn  n ) )
14 sbceq1a 3173 . . . . . 6  |-  ( f  =  g  ->  ( ph 
<-> 
[. g  /  f ]. ph ) )
1514, 5syl6bbr 256 . . . . 5  |-  ( f  =  g  ->  ( ph 
<->  ph' ) )
16 sbceq1a 3173 . . . . . 6  |-  ( f  =  g  ->  ( ps 
<-> 
[. g  /  f ]. ps ) )
1716, 8syl6bbr 256 . . . . 5  |-  ( f  =  g  ->  ( ps 
<->  ps' ) )
1813, 15, 173anbi123d 1255 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
1918rexbidv 2728 . . 3  |-  ( f  =  g  ->  ( E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
202, 12, 19cbvab 2556 . 2  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
211, 20eqtri 2458 1  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ w3a 937    = wceq 1653   {cab 2424   E.wrex 2708   [.wsbc 3163    Fn wfn 5451
This theorem is referenced by:  bnj849  29358  bnj893  29361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-fun 5458  df-fn 5459
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