Theorem modal-5 1638

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

Assertion

Ref	Expression
modal-5	⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem modal-5

Step	Hyp	Ref	Expression
1		hbn1 1630	1 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337

This theorem is referenced by: (None)