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Theorem tfrlem3-2d 5869
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
Hypothesis
Ref Expression
tfrlem3-2d.1 (φx(Fun 𝐹 (𝐹x) V))
Assertion
Ref Expression
tfrlem3-2d (φ → (Fun 𝐹 (𝐹g) V))
Distinct variable group:   x,g,𝐹
Allowed substitution hints:   φ(x,g)

Proof of Theorem tfrlem3-2d
StepHypRef Expression
1 tfrlem3-2d.1 . . 3 (φx(Fun 𝐹 (𝐹x) V))
2 fveq2 5121 . . . . . 6 (x = g → (𝐹x) = (𝐹g))
32eleq1d 2103 . . . . 5 (x = g → ((𝐹x) V ↔ (𝐹g) V))
43anbi2d 437 . . . 4 (x = g → ((Fun 𝐹 (𝐹x) V) ↔ (Fun 𝐹 (𝐹g) V)))
54cbvalv 1791 . . 3 (x(Fun 𝐹 (𝐹x) V) ↔ g(Fun 𝐹 (𝐹g) V))
61, 5sylib 127 . 2 (φg(Fun 𝐹 (𝐹g) V))
7619.21bi 1447 1 (φ → (Fun 𝐹 (𝐹g) V))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   wcel 1390  Vcvv 2551  Fun wfun 4839  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by:  tfrlemisucfn  5879  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemibfn  5883  tfrlemi14d  5888
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