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Theorem bj-hbxfrbi 33860
Description: Closed form of hbxfrbi 1816. Note: it is less important than nfbiit 1842. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 33967) in order not to require sp 2172 (modal T). See bj-hbyfrbi 33861 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-hbxfrbi (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Proof of Theorem bj-hbxfrbi
StepHypRef Expression
1 simpl 483 . 2 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (𝜑𝜓))
2 albi 1810 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
32adantl 482 . 2 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
41, 3imbi12d 346 1 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397
This theorem is referenced by:  bj-nnfbi  33954
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