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Theorem bnj873 29807
Description: Technical lemma for bnj69 29891. 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 1769 . . 3  |-  F/ g E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
3 nfcv 2612 . . . 4  |-  F/_ f D
4 nfv 1769 . . . . 5  |-  F/ f  g  Fn  n
5 bnj873.7 . . . . . 6  |-  ( ph'  <->  [. g  /  f ]. ph )
6 nfsbc1v 3275 . . . . . 6  |-  F/ f
[. g  /  f ]. ph
75, 6nfxfr 1704 . . . . 5  |-  F/ f ph'
8 bnj873.8 . . . . . 6  |-  ( ps'  <->  [. g  /  f ]. ps )
9 nfsbc1v 3275 . . . . . 6  |-  F/ f
[. g  /  f ]. ps
108, 9nfxfr 1704 . . . . 5  |-  F/ f ps'
114, 7, 10nf3an 2033 . . . 4  |-  F/ f ( g  Fn  n  /\  ph'  /\  ps' )
123, 11nfrex 2848 . . 3  |-  F/ f E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' )
13 fneq1 5674 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  n  <->  g  Fn  n ) )
14 sbceq1a 3266 . . . . . 6  |-  ( f  =  g  ->  ( ph 
<-> 
[. g  /  f ]. ph ) )
1514, 5syl6bbr 271 . . . . 5  |-  ( f  =  g  ->  ( ph 
<->  ph' ) )
16 sbceq1a 3266 . . . . . 6  |-  ( f  =  g  ->  ( ps 
<-> 
[. g  /  f ]. ps ) )
1716, 8syl6bbr 271 . . . . 5  |-  ( f  =  g  ->  ( ps 
<->  ps' ) )
1813, 15, 173anbi123d 1365 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
1918rexbidv 2892 . . 3  |-  ( f  =  g  ->  ( E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
202, 12, 19cbvab 2594 . 2  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
211, 20eqtri 2493 1  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ w3a 1007    = wceq 1452   {cab 2457   E.wrex 2757   [.wsbc 3255    Fn wfn 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5591  df-fn 5592
This theorem is referenced by:  bnj849  29808  bnj893  29811
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