Theorem 3impexpbicomVD 39092

Description: Virtual deduction proof of 3impexpbicom 38685. 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 <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
2::	|- ( ( th <-> ta ) <-> ( ta <-> th ) )
3:1,2,?: e10 38919	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
4:3,?: e1a 38852	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
5:4:	|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
6::	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
7:6,?: e1a 38852	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
8:7,2,?: e10 38919	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
9:8:	|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) )
qed:5,9,?: e00 38995	|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

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

Assertion

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

Proof of Theorem 3impexpbicomVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 |- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
2		bicom 212	. . . . 5 |- ( ( th <-> ta ) <-> ( ta <-> th ) )
3		imbi2 338	. . . . . 6 |- ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) )
4	3	biimpcd 239	. . . . 5 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) )
5	1, 2, 4	e10 38919	. . . 4 |- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
6		3impexp 1289	. . . . 5 |- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
7	6	biimpi 206	. . . 4 |- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
8	5, 7	e1a 38852	. . 3 |- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
9	8	in1 38787	. 2 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
10		idn1 38790	. . . . 5 |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
11	6	biimpri 218	. . . . 5 |- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) )
12	10, 11	e1a 38852	. . . 4 |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
13	3	biimprcd 240	. . . 4 |- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) )
14	12, 2, 13	e10 38919	. . 3 |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
15	14	in1 38787	. 2 |- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) )
16		impbi 198	. 2 |- ( ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) -> ( ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) ) )
17	9, 15, 16	e00 38995	1 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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)