Theorem bj-modalbe 37682

Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2349. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-modalbe	⊢ (𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem bj-modalbe

Step	Hyp	Ref	Expression
1		modal-b 2349	. 2 ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2		df-ex 1812	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
3	2	biimpri 231	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
4	1, 3	sylg 1855	1 ⊢ (𝜑 → ∀𝑥∃𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bj-modal4  37710  bj-19.12  37717