Theorem modal-b 2337

Description: The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)

Assertion

Ref	Expression
modal-b	⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem modal-b

Step	Hyp	Ref	Expression
1		axc7 2335	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝜑 → ¬ 𝜑)
2	1	con4i 114	1 ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by:  bj-modalbe  34026