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Theorem bj-nfbi 31793
 Description: Closed form of nfbii 1770 (with df-bj-nf 31765 instead of nf5 2102, which would require more axioms). (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-nfbi (∀𝑥(𝜑𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓))

Proof of Theorem bj-nfbi
StepHypRef Expression
1 exbi 1762 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2 albi 1736 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
31, 2imbi12d 333 . 2 (∀𝑥(𝜑𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
4 df-bj-nf 31765 . 2 (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5 df-bj-nf 31765 . 2 (ℲℲ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
63, 4, 53bitr4g 302 1 (∀𝑥(𝜑𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  ℲℲwnff 31764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-bj-nf 31765 This theorem is referenced by:  bj-nfxfr  31794
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