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Theorem hbxfreq 2144
 Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1361 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 2104 . 2 (𝑦𝐴𝑦𝐵)
3 hbxfr.2 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
42, 3hbxfrbi 1361 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393 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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by: (None)
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