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Theorem tfrlem3-2 5927
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.)
Hypothesis
Ref Expression
tfrlem3-2.1  |-  ( Fun 
F  /\  ( F `  x )  e.  _V )
Assertion
Ref Expression
tfrlem3-2  |-  ( Fun 
F  /\  ( F `  g )  e.  _V )
Distinct variable group:    x, g, F

Proof of Theorem tfrlem3-2
StepHypRef Expression
1 fveq2 5178 . . . 4  |-  ( x  =  g  ->  ( F `  x )  =  ( F `  g ) )
21eleq1d 2106 . . 3  |-  ( x  =  g  ->  (
( F `  x
)  e.  _V  <->  ( F `  g )  e.  _V ) )
32anbi2d 437 . 2  |-  ( x  =  g  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  g )  e.  _V ) ) )
4 tfrlem3-2.1 . 2  |-  ( Fun 
F  /\  ( F `  x )  e.  _V )
53, 4chvarv 1812 1  |-  ( Fun 
F  /\  ( F `  g )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557   Fun wfun 4896   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910
This theorem is referenced by: (None)
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