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Theorem bj-hbxfrbi 32285
Description: Closed form of hbxfrbi 1749. Notes: it is less important than nfbiit 1774; it requires sp 2051 (unlike nfbiit 1774); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-hbxfrbi (∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Proof of Theorem bj-hbxfrbi
StepHypRef Expression
1 sp 2051 . 2 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
2 albi 1743 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
31, 2imbi12d 334 1 (∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
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  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by: (None)
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