Theorem bj-hbntbi 36090

Description: Strengthening hbnt 2282 by replacing its consequent with a biconditional. See also hbntg 35310 and hbntal 43890. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36089. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbntbi	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Proof of Theorem bj-hbntbi

Step	Hyp	Ref	Expression
1		bj-19.9htbi 36089	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
2	1	bicomd 222	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
3	2	notbid 318	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4		alnex 1775	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	3, 4	bitr4di 289	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1531  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)