Theorem bj-hbntbi 36670

Description: Strengthening hbnt 2298 by replacing its consequent with a biconditional. See also hbntg 35769 and hbntal 44524. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36669. (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 36669	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
2	1	bicomd 223	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
3	2	notbid 318	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4		alnex 1779	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	3, 4	bitr4di 289	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by: (None)