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Theorem bnj873 29272
Description: Technical lemma for bnj69 29356. 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 1609 . . 3  |-  F/ g E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
3 nfcv 2432 . . . 4  |-  F/_ f D
4 nfv 1609 . . . . 5  |-  F/ f  g  Fn  n
5 bnj873.7 . . . . . 6  |-  ( ph'  <->  [. g  /  f ]. ph )
6 nfcv 2432 . . . . . . 7  |-  F/_ f
g
76nfsbc1 3022 . . . . . 6  |-  F/ f
[. g  /  f ]. ph
85, 7nfxfr 1560 . . . . 5  |-  F/ f ph'
9 bnj873.8 . . . . . 6  |-  ( ps'  <->  [. g  /  f ]. ps )
106nfsbc1 3022 . . . . . 6  |-  F/ f
[. g  /  f ]. ps
119, 10nfxfr 1560 . . . . 5  |-  F/ f ps'
124, 8, 11nf3an 1786 . . . 4  |-  F/ f ( g  Fn  n  /\  ph'  /\  ps' )
133, 12nfrex 2611 . . 3  |-  F/ f E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' )
14 fneq1 5349 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  n  <->  g  Fn  n ) )
15 sbceq1a 3014 . . . . . 6  |-  ( f  =  g  ->  ( ph 
<-> 
[. g  /  f ]. ph ) )
1615, 5syl6bbr 254 . . . . 5  |-  ( f  =  g  ->  ( ph 
<->  ph' ) )
17 sbceq1a 3014 . . . . . 6  |-  ( f  =  g  ->  ( ps 
<-> 
[. g  /  f ]. ps ) )
1817, 9syl6bbr 254 . . . . 5  |-  ( f  =  g  ->  ( ps 
<->  ps' ) )
1914, 16, 183anbi123d 1252 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
2019rexbidv 2577 . . 3  |-  ( f  =  g  ->  ( E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' ) ) )
212, 13, 20cbvab 2414 . 2  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
221, 21eqtri 2316 1  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    = wceq 1632   {cab 2282   E.wrex 2557   [.wsbc 3004    Fn wfn 5266
This theorem is referenced by:  bnj849  29273  bnj893  29276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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