Theorem modal-b 2355

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 2353	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝜑 → ¬ 𝜑)
2	1	con4i 115	1 ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-modalbe  37354