Theorem bj-nf2 31766

Description: Alternate definition of df-bj-nf 31765. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
bj-nf2	⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Proof of Theorem bj-nf2

Step	Hyp	Ref	Expression
1		df-bj-nf 31765	. 2 ⊢ (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		imor 427	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
3		orcom 401	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 2, 3	3bitri 285	1 ⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  ∀wal 1473  ∃wex 1695  ℲℲwnff 31764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384  df-bj-nf 31765

This theorem is referenced by:  bj-nf3  31767  bj-nfntht  31770