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Theorem hbxfrbi 1749
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2727 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 1744 . 2 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
41, 2, 33imtr4i 281 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  hbn1fw  1969  hbe1w  1973  hbe1  2018  hbexOLD  2149  hbab1  2610  hbab  2612  hbxfreq  2727  hbral  2938  bnj982  30554  bnj1095  30557  bnj1096  30558  bnj1276  30590  bnj594  30687  bnj1445  30817  bj-hbab1  32411  hbra2VD  38576
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