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Theorem hbxfreq 2717
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1742 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 2680 . 2 (𝑦𝐴𝑦𝐵)
3 hbxfr.2 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
42, 3hbxfrbi 1742 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473   = wceq 1475  wcel 1977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606
This theorem is referenced by:  bnj1317  29940  bnj1441  29959  bnj1309  30138
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