Theorem hbntal 38769

Description: A closed form of hbn 2146. hbnt 2144 is another closed form of hbn 2146. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbntal	|- ( A. x ( ph -> A. x ph ) -> A. x ( -. ph -> A. x -. ph ) )

Proof of Theorem hbntal

Step	Hyp	Ref	Expression
1		hba1 2151	. 2 |- ( A. x ( ph -> A. x ph ) -> A. x A. x ( ph -> A. x ph ) )
2		axc7 2132	. . . . 5 |- ( -. A. x -. A. x ph -> ph )
3	2	con1i 144	. . . 4 |- ( -. ph -> A. x -. A. x ph )
4		con3 149	. . . . 5 |- ( ( ph -> A. x ph ) -> ( -. A. x ph -> -. ph ) )
5	4	al2imi 1743	. . . 4 |- ( A. x ( ph -> A. x ph ) -> ( A. x -. A. x ph -> A. x -. ph ) )
6	3, 5	syl5 34	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )
7	6	alimi 1739	. 2 |- ( A. x A. x ( ph -> A. x ph ) -> A. x ( -. ph -> A. x -. ph ) )
8	1, 7	syl 17	1 |- ( A. x ( ph -> A. x ph ) -> A. x ( -. ph -> A. x -. ph ) )

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

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-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  hbimpg  38770  hbimpgVD  39140  hbexgVD  39142