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Theorem hbxfrbi 1897
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2864 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 1892 . 2 (∀𝑥𝜑 ↔ ∀𝑥𝜓)
41, 2, 33imtr4i 281 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  hbn1fw  2119  hbe1w  2123  hbe1  2166  hbexOLD  2295  hbab1  2745  hbab  2747  hbxfreq  2864  hbral  3077  bnj982  31152  bnj1095  31155  bnj1096  31156  bnj1276  31188  bnj594  31285  bnj1445  31415  bj-hbab1  33073  hbra2VD  39591
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