Theorem bj-hbntbi 32695

Description: Strengthening hbnt 2144 by replacing its succedent with a biconditional. See also hbntg 31711 and hbntal 38769. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 32694. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbntbi	|- ( A. x ( ph -> A. x ph ) -> ( -. ph <-> A. x -. ph ) )

Proof of Theorem bj-hbntbi

Step	Hyp	Ref	Expression
1		bj-19.9htbi 32694	. . . 4 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph <-> ph ) )
2	1	bicomd 213	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( ph <-> E. x ph ) )
3	2	notbid 308	. 2 |- ( A. x ( ph -> A. x ph ) -> ( -. ph <-> -. E. x ph ) )
4		alnex 1706	. 2 |- ( A. x -. ph <-> -. E. x ph )
5	3, 4	syl6bbr 278	1 |- ( A. x ( ph -> A. x ph ) -> ( -. ph <-> A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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)