ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hbxfreq GIF version

Theorem hbxfreq 2195
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1407 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 2155 . 2 (𝑦𝐴𝑦𝐵)
3 hbxfr.2 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
42, 3hbxfrbi 1407 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1288   = wceq 1290  wcel 1439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator