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Theorem tfrlem3-2d 5958
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
Hypothesis
Ref Expression
tfrlem3-2d.1 (𝜑 → ∀𝑥(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 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
2 fveq2 5205 . . . . . 6 (𝑥 = 𝑔 → (𝐹𝑥) = (𝐹𝑔))
32eleq1d 2122 . . . . 5 (𝑥 = 𝑔 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑔) ∈ V))
43anbi2d 445 . . . 4 (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑔) ∈ V)))
54cbvalv 1810 . . 3 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
61, 5sylib 131 . 2 (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
7619.21bi 1466 1 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wcel 1409  Vcvv 2574  Fun wfun 4923  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937
This theorem is referenced by:  tfrlemisucfn  5968  tfrlemisucaccv  5969  tfrlemibxssdm  5971  tfrlemibfn  5972  tfrlemi14d  5977
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