ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem3-2d Unicode version

Theorem tfrlem3-2d 6091
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
Hypothesis
Ref Expression
tfrlem3-2d.1  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
Assertion
Ref Expression
tfrlem3-2d  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Distinct variable group:    x, g, F
Allowed substitution hints:    ph( x, g)

Proof of Theorem tfrlem3-2d
StepHypRef Expression
1 tfrlem3-2d.1 . . 3  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
2 fveq2 5318 . . . . . 6  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
32eleq1d 2157 . . . . 5  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
43anbi2d 453 . . . 4  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
54cbvalv 1843 . . 3  |-  ( A. x ( Fun  F  /\  ( F `  x
)  e.  _V )  <->  A. g ( Fun  F  /\  ( F `  g
)  e.  _V )
)
61, 5sylib 121 . 2  |-  ( ph  ->  A. g ( Fun 
F  /\  ( F `  g )  e.  _V ) )
7619.21bi 1496 1  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1288    e. wcel 1439   _Vcvv 2620   Fun wfun 5022   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036
This theorem is referenced by:  tfrlemisucfn  6103  tfrlemisucaccv  6104  tfrlemibxssdm  6106  tfrlemibfn  6107  tfrlemi14d  6112
  Copyright terms: Public domain W3C validator