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Metamath Proof Explorer

Theorem hbimtg

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

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

Proof

Step Hyp Ref Expression
1 hbntg 
 |-  ( A. x ( ph -> A. x ch ) -> ( -. ch -> A. x -. ph ) )
2 pm2.21 
 |-  ( -. ph -> ( ph -> th ) )
3 2 alimi 
 |-  ( A. x -. ph -> A. x ( ph -> th ) )
4 1 3 syl6 
 |-  ( A. x ( ph -> A. x ch ) -> ( -. ch -> A. x ( ph -> th ) ) )
5 4 adantr 
 |-  ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( -. ch -> A. x ( ph -> th ) ) )
6 ala1 
 |-  ( A. x th -> A. x ( ph -> th ) )
7 6 imim2i 
 |-  ( ( ps -> A. x th ) -> ( ps -> A. x ( ph -> th ) ) )
8 7 adantl 
 |-  ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( ps -> A. x ( ph -> th ) ) )
9 5 8 jad 
 |-  ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> ( ( ch -> ps ) -> A. x ( ph -> th ) ) )