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Theorem bnj124 32146
Description: Technical lemma for bnj150 32151. 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 3330 . . 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 32139 . . . 4  |-  F  e. 
_V
6 nfv 1674 . . . . 5  |-  F/ f ( R  FrSe  A  /\  x  e.  A
)
76sbc19.21g 3343 . . . 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 5 . . 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 5583 . . . . . . . 8  |-  ( f  =  z  ->  (
f  Fn  1o  <->  z  Fn  1o ) )
10 fneq1 5583 . . . . . . . 8  |-  ( z  =  F  ->  (
z  Fn  1o  <->  F  Fn  1o ) )
119, 10sbcie2g 3304 . . . . . . 7  |-  ( F  e.  _V  ->  ( [. F  /  f ]. f  Fn  1o  <->  F  Fn  1o ) )
125, 11ax-mp 5 . . . . . 6  |-  ( [. F  /  f ]. f  Fn  1o  <->  F  Fn  1o )
1312bicomi 202 . . . . 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 32004 . . . 4  |-  ( [. F  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( F  Fn  1o  /\  ph"  /\  ps" ) )
1716imbi2i 312 . . 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 271 . 2  |-  ( [. F  /  f ]. ze'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
191, 18bitri 249 1  |-  ( ze"  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   _Vcvv 3054   [.wsbc 3270   (/)c0 3721   {csn 3961   <.cop 3967    Fn wfn 5497   1oc1o 6999    predc-bnj14 31958    FrSe w-bnj15 31962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-fun 5504  df-fn 5505
This theorem is referenced by:  bnj150  32151  bnj153  32155
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