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

Theorem tfrlem3-2d 6272
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 5481 . . . . . 6  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
32eleq1d 2233 . . . . 5  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
43anbi2d 460 . . . 4  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
54cbvalv 1904 . . 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 1545 1  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1340    e. wcel 2135   _Vcvv 2722   Fun wfun 5177   ` cfv 5183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2724  df-un 3116  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-iota 5148  df-fv 5191
This theorem is referenced by:  tfrlemisucfn  6284  tfrlemisucaccv  6285  tfrlemibxssdm  6287  tfrlemibfn  6288  tfrlemi14d  6293
  Copyright terms: Public domain W3C validator