Theorem exbirVD 39088

Description: Virtual deduction proof of exbir 38684. 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 <-> th ) ) ,. ( ph /\ ps ) ->. ( ph /\ ps ) ).
3::	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) , th ->. th ).
5:1,2,?: e12 38951	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( ch <-> th ) ).
6:3,5,?: e32 38985	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) , th ->. ch ).
7:6:	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( th -> ch ) ).
8:7:	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( th -> ch ) ) ).
9:8,?: e1a 38852	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ph -> ( ps -> ( th -> ch ) ) ) ).
qed:9:	|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )

(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem exbirVD

Step	Hyp	Ref	Expression
1		idn3 38840	. . . . . 6 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ,. th ->. th ).
2		idn1 38790	. . . . . . 7 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( ch <-> th ) ) ).
3		idn2 38838	. . . . . . 7 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( ph /\ ps ) ).
4		id 22	. . . . . . 7 |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ( ph /\ ps ) -> ( ch <-> th ) ) )
5	2, 3, 4	e12 38951	. . . . . 6 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( ch <-> th ) ).
6		biimpr 210	. . . . . . 7 |- ( ( ch <-> th ) -> ( th -> ch ) )
7	6	com12 32	. . . . . 6 |- ( th -> ( ( ch <-> th ) -> ch ) )
8	1, 5, 7	e32 38985	. . . . 5 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ,. th ->. ch ).
9	8	in3 38834	. . . 4 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( th -> ch ) ).
10	9	in2 38830	. . 3 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( th -> ch ) ) ).
11		pm3.3 460	. . 3 |- ( ( ( ph /\ ps ) -> ( th -> ch ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
12	10, 11	e1a 38852	. 2 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ph -> ( ps -> ( th -> ch ) ) ) ).
13	12	in1 38787	1 |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )

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

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  df-vd2 38794  df-vd3 38806

This theorem is referenced by: (None)