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Theorem hbxfrbi 1919
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2873 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
hbxfrbi.1 (𝜑𝜓)
hbxfrbi.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
hbxfrbi (𝜑 → ∀𝑥𝜑)

Proof of Theorem hbxfrbi
StepHypRef Expression
1 hbxfrbi.2 . 2 (𝜓 → ∀𝑥𝜓)
2 hbxfrbi.1 . 2 (𝜑𝜓)
32albii 1914 . 2 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
41, 2, 33imtr4i 283 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198
This theorem is referenced by:  hbn1fw  2140  hbe1w  2143  hbe1  2185  hbab1  2754  hbab  2756  hbxfreq  2873  hbral  3090  bnj982  31297  bnj1095  31300  bnj1096  31301  bnj1276  31333  bnj594  31430  bnj1445  31560  bj-hbab1  33201  hbra2VD  39748
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