Theorem 3impexpVD 39091

Description: Virtual deduction proof of 3impexp 1289. 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 38919	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
4:3,?: e1a 38852	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
5:4,?: e1a 38852	|- (. ( ( 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 38852	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
9:8,?: e1a 38852	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
10:2,9,?: e01 38916	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
11:10:	|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
qed:6,11,?: e00 38995	|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 3impexpVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . 6 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
2		df-3an 1039	. . . . . 6 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
3		imbi1 337	. . . . . . 7 |- ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ( ( ph /\ ps ) /\ ch ) -> th ) ) )
4	3	biimpcd 239	. . . . . 6 |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ph /\ ps ) /\ ch ) -> th ) ) )
5	1, 2, 4	e10 38919	. . . . 5 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
6		pm3.3 460	. . . . 5 |- ( ( ( ( ph /\ ps ) /\ ch ) -> th ) -> ( ( ph /\ ps ) -> ( ch -> th ) ) )
7	5, 6	e1a 38852	. . . 4 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
8		pm3.3 460	. . . 4 |- ( ( ( ph /\ ps ) -> ( ch -> th ) ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
9	7, 8	e1a 38852	. . 3 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
10	9	in1 38787	. 2 |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
11		idn1 38790	. . . . . 6 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
12		pm3.31 461	. . . . . 6 |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps ) -> ( ch -> th ) ) )
13	11, 12	e1a 38852	. . . . 5 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
14		pm3.31 461	. . . . 5 |- ( ( ( ph /\ ps ) -> ( ch -> th ) ) -> ( ( ( ph /\ ps ) /\ ch ) -> th ) )
15	13, 14	e1a 38852	. . . 4 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
16	3	biimprd 238	. . . 4 |- ( ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) -> ( ( ( ( ph /\ ps ) /\ ch ) -> th ) -> ( ( ph /\ ps /\ ch ) -> th ) ) )
17	2, 15, 16	e01 38916	. . 3 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
18	17	in1 38787	. 2 |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
19		impbi 198	. 2 |- ( ( ( ( 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 38995	1 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-vd1 38786

This theorem is referenced by: (None)