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Theorem tfrlem3-2d 6033
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 5270 . . . . . 6 (𝑥 = 𝑔 → (𝐹𝑥) = (𝐹𝑔))
32eleq1d 2153 . . . . 5 (𝑥 = 𝑔 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑔) ∈ V))
43anbi2d 452 . . . 4 (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑔) ∈ V)))
54cbvalv 1839 . . 3 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
61, 5sylib 120 . 2 (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
7619.21bi 1493 1 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285  wcel 1436  Vcvv 2615  Fun wfun 4977  cfv 4983
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-iota 4948  df-fv 4991
This theorem is referenced by:  tfrlemisucfn  6045  tfrlemisucaccv  6046  tfrlemibxssdm  6048  tfrlemibfn  6049  tfrlemi14d  6054
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