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Theorem bnj873 33070
Description: Technical lemma for bnj69 33154. 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 1683 . . 3  |-  F/ g E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
3 nfcv 2629 . . . 4  |-  F/_ f D
4 nfv 1683 . . . . 5  |-  F/ f  g  Fn  n
5 bnj873.7 . . . . . 6  |-  ( ph'  <->  [. g  /  f ]. ph )
6 nfsbc1v 3351 . . . . . 6  |-  F/ f
[. g  /  f ]. ph
75, 6nfxfr 1625 . . . . 5  |-  F/ f ph'
8 bnj873.8 . . . . . 6  |-  ( ps'  <->  [. g  /  f ]. ps )
9 nfsbc1v 3351 . . . . . 6  |-  F/ f
[. g  /  f ]. ps
108, 9nfxfr 1625 . . . . 5  |-  F/ f ps'
114, 7, 10nf3an 1877 . . . 4  |-  F/ f ( g  Fn  n  /\  ph'  /\  ps' )
123, 11nfrex 2927 . . 3  |-  F/ f E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' )
13 fneq1 5668 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  n  <->  g  Fn  n ) )
14 sbceq1a 3342 . . . . . 6  |-  ( f  =  g  ->  ( ph 
<-> 
[. g  /  f ]. ph ) )
1514, 5syl6bbr 263 . . . . 5  |-  ( f  =  g  ->  ( ph 
<->  ph' ) )
16 sbceq1a 3342 . . . . . 6  |-  ( f  =  g  ->  ( ps 
<-> 
[. g  /  f ]. ps ) )
1716, 8syl6bbr 263 . . . . 5  |-  ( f  =  g  ->  ( ps 
<->  ps' ) )
1813, 15, 173anbi123d 1299 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
1918rexbidv 2973 . . 3  |-  ( f  =  g  ->  ( E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
202, 12, 19cbvab 2608 . 2  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
211, 20eqtri 2496 1  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 973    = wceq 1379   {cab 2452   E.wrex 2815   [.wsbc 3331    Fn wfn 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5589  df-fn 5590
This theorem is referenced by:  bnj849  33071  bnj893  33074
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