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Theorem bnj124 28581
Description: Technical lemma for bnj150 28586. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj124.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
bnj124.2  |-  ( ph"  <->  [. F  / 
f ]. ph' )
bnj124.3  |-  ( ps"  <->  [. F  / 
f ]. ps' )
bnj124.4  |-  ( ze"  <->  [. F  / 
f ]. ze' )
bnj124.5  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Assertion
Ref Expression
bnj124  |-  ( ze"  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
Distinct variable groups:    A, f    R, f    x, f
Allowed substitution hints:    A( x)    R( x)    F( x, f)    ph'( x, f)    ps'( x, f)    ze'( x, f)    ph"( x, f)    ps"( x, f)    ze"( x, f)

Proof of Theorem bnj124
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj124.4 . 2  |-  ( ze"  <->  [. F  / 
f ]. ze' )
2 bnj124.5 . . . 4  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
32sbcbii 3160 . . 3  |-  ( [. F  /  f ]. ze'  <->  [. F  / 
f ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ph' 
/\  ps' ) ) )
4 bnj124.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54bnj95 28574 . . . 4  |-  F  e. 
_V
6 nfv 1626 . . . . 5  |-  F/ f ( R  FrSe  A  /\  x  e.  A
)
76sbc19.21g 3169 . . . 4  |-  ( F  e.  _V  ->  ( [. F  /  f ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  [. F  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) ) ) )
85, 7ax-mp 8 . . 3  |-  ( [. F  /  f ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  [. F  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
9 fneq1 5475 . . . . . . . 8  |-  ( f  =  z  ->  (
f  Fn  1o  <->  z  Fn  1o ) )
10 fneq1 5475 . . . . . . . 8  |-  ( z  =  F  ->  (
z  Fn  1o  <->  F  Fn  1o ) )
119, 10sbcie2g 3138 . . . . . . 7  |-  ( F  e.  _V  ->  ( [. F  /  f ]. f  Fn  1o  <->  F  Fn  1o ) )
125, 11ax-mp 8 . . . . . 6  |-  ( [. F  /  f ]. f  Fn  1o  <->  F  Fn  1o )
1312bicomi 194 . . . . 5  |-  ( F  Fn  1o  <->  [. F  / 
f ]. f  Fn  1o )
14 bnj124.2 . . . . 5  |-  ( ph"  <->  [. F  / 
f ]. ph' )
15 bnj124.3 . . . . 5  |-  ( ps"  <->  [. F  / 
f ]. ps' )
1613, 14, 15, 5bnj206 28437 . . . 4  |-  ( [. F  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( F  Fn  1o  /\  ph"  /\  ps" ) )
1716imbi2i 304 . . 3  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  [. F  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
183, 8, 173bitri 263 . 2  |-  ( [. F  /  f ]. ze'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
191, 18bitri 241 1  |-  ( ze"  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900   [.wsbc 3105   (/)c0 3572   {csn 3758   <.cop 3761    Fn wfn 5390   1oc1o 6654    predc-bnj14 28391    FrSe w-bnj15 28395
This theorem is referenced by:  bnj150  28586  bnj153  28590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-fun 5397  df-fn 5398
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