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Theorem tfrlem3-2d 5961
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 5209 . . . . . 6  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
32eleq1d 2148 . . . . 5  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
43anbi2d 452 . . . 4  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
54cbvalv 1836 . . 3  |-  ( A. x ( Fun  F  /\  ( F `  x
)  e.  _V )  <->  A. g ( Fun  F  /\  ( F `  g
)  e.  _V )
)
61, 5sylib 120 . 2  |-  ( ph  ->  A. g ( Fun 
F  /\  ( F `  g )  e.  _V ) )
7619.21bi 1491 1  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    e. wcel 1434   _Vcvv 2602   Fun wfun 4926   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940
This theorem is referenced by:  tfrlemisucfn  5973  tfrlemisucaccv  5974  tfrlemibxssdm  5976  tfrlemibfn  5977  tfrlemi14d  5982
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