Metamath Proof Explorer

Theorem 3impexpVD

Description: Virtual deduction proof of 3impexp . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1:: |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
2:: |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
3:1,2,?: e10 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
4:3,?: e1a |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
5:4,?: e1a |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
6:5: |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
7:: |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
8:7,?: e1a |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
9:8,?: e1a |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
10:2,9,?: e01 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
11:10: |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
qed:6,11,?: e00 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3impexpVD 
|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
2 df-3an 
 |-  ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
3 imbi1 
 |-  ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ( ( ph /\ ps ) /\ ch ) -> th ) ) )
4 3 biimpcd 
 |-  ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ph /\ ps ) /\ ch ) -> th ) ) )
5 1 2 4 e10 
 |-  (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
6 pm3.3 
 |-  ( ( ( ( ph /\ ps ) /\ ch ) -> th ) -> ( ( ph /\ ps ) -> ( ch -> th ) ) )
7 5 6 e1a 
 |-  (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
8 pm3.3 
 |-  ( ( ( ph /\ ps ) -> ( ch -> th ) ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
9 7 8 e1a 
 |-  (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
10 9 in1 
 |-  ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
11 idn1 
 |-  (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
12 pm3.31 
 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps ) -> ( ch -> th ) ) )
13 11 12 e1a 
 |-  (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
14 pm3.31 
 |-  ( ( ( ph /\ ps ) -> ( ch -> th ) ) -> ( ( ( ph /\ ps ) /\ ch ) -> th ) )
15 13 14 e1a 
 |-  (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
16 3 biimprd 
 |-  ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ( ph /\ ps ) /\ ch ) -> th ) -> ( ( ph /\ ps /\ ch ) -> th ) ) )
17 2 15 16 e01 
 |-  (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
18 17 in1 
 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
19 impbi 
 |-  ( ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) ) -> ( ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) ) -> ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) ) ) )
20 10 18 19 e00 
 |-  ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )