Theorem bj-modalb 34584

Description: A short form of the axiom B of modal logic using only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-modalb

Step	Hyp	Ref	Expression
1		axc7 2318	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
2	1	con1i 149	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by: (None)