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Theorem tfrlem3-2d 6332
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 5531 . . . . . 6  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
32eleq1d 2258 . . . . 5  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
43anbi2d 464 . . . 4  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
54cbvalv 1929 . . 3  |-  ( A. x ( Fun  F  /\  ( F `  x
)  e.  _V )  <->  A. g ( Fun  F  /\  ( F `  g
)  e.  _V )
)
61, 5sylib 122 . 2  |-  ( ph  ->  A. g ( Fun 
F  /\  ( F `  g )  e.  _V ) )
7619.21bi 1569 1  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362    e. wcel 2160   _Vcvv 2752   Fun wfun 5226   ` cfv 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5240
This theorem is referenced by:  tfrlemisucfn  6344  tfrlemisucaccv  6345  tfrlemibxssdm  6347  tfrlemibfn  6348  tfrlemi14d  6353
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