Theorem bj-modalbe 32678

Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2142. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-modalbe	|- ( ph -> A. x E. x ph )

Proof of Theorem bj-modalbe

Step	Hyp	Ref	Expression
1		modal-b 2142	. 2 |- ( ph -> A. x -. A. x -. ph )
2		df-ex 1705	. . 3 |- ( E. x ph <-> -. A. x -. ph )
3	2	biimpri 218	. 2 |- ( -. A. x -. ph -> E. x ph )
4	1, 3	sylg 1750	1 |- ( ph -> A. x E. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)