Theorem hbimtg 31712

Description: A more general and closed form of hbim 2127. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
hbimtg	|- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( ( ch -> ps ) -> A. x ( ph -> th ) ) )

Proof of Theorem hbimtg

Step	Hyp	Ref	Expression
1		hbntg 31711	. . . 4 |- ( A. x ( ph -> A. x ch ) -> ( -. ch -> A. x -. ph ) )
2		pm2.21 120	. . . . 5 |- ( -. ph -> ( ph -> th ) )
3	2	alimi 1739	. . . 4 |- ( A. x -. ph -> A. x ( ph -> th ) )
4	1, 3	syl6 35	. . 3 |- ( A. x ( ph -> A. x ch ) -> ( -. ch -> A. x ( ph -> th ) ) )
5	4	adantr 481	. 2 |- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( -. ch -> A. x ( ph -> th ) ) )
6		ala1 1741	. . . 4 |- ( A. x th -> A. x ( ph -> th ) )
7	6	imim2i 16	. . 3 |- ( ( ps -> A. x th ) -> ( ps -> A. x ( ph -> th ) ) )
8	7	adantl 482	. 2 |- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( ps -> A. x ( ph -> th ) ) )
9	5, 8	jad 174	1 |- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( ( ch -> ps ) -> A. x ( ph -> th ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   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-an 386  df-ex 1705

This theorem is referenced by:  hbimg  31715