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Theorem hbxfreq 2224
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1433 for equivalence version. (Contributed by NM, 21-Aug-2007.)
Hypotheses
Ref Expression
hbxfr.1 𝐴 = 𝐵
hbxfr.2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Assertion
Ref Expression
hbxfreq (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Proof of Theorem hbxfreq
StepHypRef Expression
1 hbxfr.1 . . 3 𝐴 = 𝐵
21eleq2i 2184 . 2 (𝑦𝐴𝑦𝐵)
3 hbxfr.2 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
42, 3hbxfrbi 1433 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by: (None)
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