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Theorem hbxfreq 2870
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1822 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 2831 . 2 (𝑦𝐴𝑦𝐵)
3 hbxfr.2 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
42, 3hbxfrbi 1822 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814
This theorem is referenced by:  bnj1317  34814  bnj1441  34833  bnj1441g  34834  bnj1309  35015
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