Theorem bj-modalbe 34772

Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2320. (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 2320	. 2 ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2		df-ex 1788	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
3	2	biimpri 231	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
4	1, 3	sylg 1830	1 ⊢ (𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  bj-modal4  34798  bj-19.12  34845