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Theorem nfbiit 1776
Description: Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1777. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
nfbiit (∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Proof of Theorem nfbiit
StepHypRef Expression
1 exbi 1772 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2 albi 1745 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
31, 2imbi12d 334 . 2 (∀𝑥(𝜑𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
4 df-nf 1709 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5 df-nf 1709 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
63, 4, 53bitr4g 303 1 (∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480  wex 1703  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-ex 1704  df-nf 1709
This theorem is referenced by:  nfbii  1777
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