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Theorem bnj124 29179
Description: Technical lemma for bnj150 29184. 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 3208 . . 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 29172 . . . 4  |-  F  e. 
_V
6 nfv 1629 . . . . 5  |-  F/ f ( R  FrSe  A  /\  x  e.  A
)
76sbc19.21g 3217 . . . 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 5526 . . . . . . . 8  |-  ( f  =  z  ->  (
f  Fn  1o  <->  z  Fn  1o ) )
10 fneq1 5526 . . . . . . . 8  |-  ( z  =  F  ->  (
z  Fn  1o  <->  F  Fn  1o ) )
119, 10sbcie2g 3186 . . . . . . 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 29035 . . . 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 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153   (/)c0 3620   {csn 3806   <.cop 3809    Fn wfn 5441   1oc1o 6709    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj150  29184  bnj153  29188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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