Theorem bj-nf3 31767

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

Assertion

Ref	Expression
bj-nf3	⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Proof of Theorem bj-nf3

Step	Hyp	Ref	Expression
1		bj-nf2 31766	. 2 ⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1697	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	bicomi 213	. . 3 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
4	3	orbi2i 540	. 2 ⊢ ((∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
5	1, 4	bitri 263	1 ⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   ↔ 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-ex 1696  df-bj-nf 31765

This theorem is referenced by:  bj-nf4  31768  bj-nfntht2  31771  bj-nfn  31795