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Theorem tfrlem3-2d 6327
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 5527 . . . . . 6 (𝑥 = 𝑔 → (𝐹𝑥) = (𝐹𝑔))
32eleq1d 2256 . . . . 5 (𝑥 = 𝑔 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑔) ∈ V))
43anbi2d 464 . . . 4 (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑔) ∈ V)))
54cbvalv 1927 . . 3 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
61, 5sylib 122 . 2 (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
7619.21bi 1568 1 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1361  wcel 2158  Vcvv 2749  Fun wfun 5222  cfv 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236
This theorem is referenced by:  tfrlemisucfn  6339  tfrlemisucaccv  6340  tfrlemibxssdm  6342  tfrlemibfn  6343  tfrlemi14d  6348
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