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Theorem nfbiit 1895
Description: Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1896. (Contributed by Giovanni Mascellani, 10-Apr-2018.) Reduce axiom usage. (Revised by BJ, 6-May-2019.)
Assertion
Ref Expression
nfbiit (∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Proof of Theorem nfbiit
StepHypRef Expression
1 exbi 1891 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2 albi 1862 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
31, 2imbi12d 336 . 2 (∀𝑥(𝜑𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
4 df-nf 1828 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5 df-nf 1828 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
63, 4, 53bitr4g 306 1 (∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1599  wex 1823  wnf 1827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853
This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828
This theorem is referenced by:  nfbii  1896
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