Theorem bj-modalb 31893

Description: A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.)

Assertion

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

Proof of Theorem bj-modalb

Step	Hyp	Ref	Expression
1		axc7 2117	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
2	1	con1i 143	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)