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Theorem tfrlem3-2d 6209
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 5421 . . . . . 6  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
32eleq1d 2208 . . . . 5  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
43anbi2d 459 . . . 4  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
54cbvalv 1889 . . 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 1537 1  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1329    e. wcel 1480   _Vcvv 2686   Fun wfun 5117   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131
This theorem is referenced by:  tfrlemisucfn  6221  tfrlemisucaccv  6222  tfrlemibxssdm  6224  tfrlemibfn  6225  tfrlemi14d  6230
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