Theorem imbi12VD 39109

Description: Implication form of imbi12i 340. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 336 is imbi12VD 39109 without virtual deductions and was automatically derived from imbi12VD 39109.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ph -> ch ) ).
4:1,3:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> ch ) ).
5:2,4:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> th ) ).
6:5:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) -> ( ps -> th ) ) ).
7::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ps -> th ) ).
8:1,7:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> th ) ).
9:2,8:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> ch ) ).
10:9:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ps -> th ) -> ( ph -> ch ) ) ).
11:6,10:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) <-> ( ps -> th ) ) ).
12:11:	|- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ).
qed:12:	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
imbi12VD	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Proof of Theorem imbi12VD

Step	Hyp	Ref	Expression
1		idn2 38838	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
2		idn1 38790	. . . . . . 7 |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
3		idn3 38840	. . . . . . 7 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ph -> ch ) ).
4		biimpr 210	. . . . . . . 8 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
5	4	imim1d 82	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ( ph -> ch ) -> ( ps -> ch ) ) )
6	2, 3, 5	e13 38975	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> ch ) ).
7		biimp 205	. . . . . . 7 |- ( ( ch <-> th ) -> ( ch -> th ) )
8	7	imim2d 57	. . . . . 6 |- ( ( ch <-> th ) -> ( ( ps -> ch ) -> ( ps -> th ) ) )
9	1, 6, 8	e23 38982	. . . . 5 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> th ) ).
10	9	in3 38834	. . . 4 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) -> ( ps -> th ) ) ).
11		idn3 38840	. . . . . . 7 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ps -> th ) ).
12		biimp 205	. . . . . . . 8 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
13	12	imim1d 82	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ( ps -> th ) -> ( ph -> th ) ) )
14	2, 11, 13	e13 38975	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> th ) ).
15		biimpr 210	. . . . . . 7 |- ( ( ch <-> th ) -> ( th -> ch ) )
16	15	imim2d 57	. . . . . 6 |- ( ( ch <-> th ) -> ( ( ph -> th ) -> ( ph -> ch ) ) )
17	1, 14, 16	e23 38982	. . . . 5 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> ch ) ).
18	17	in3 38834	. . . 4 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ps -> th ) -> ( ph -> ch ) ) ).
19		impbi 198	. . . 4 |- ( ( ( ph -> ch ) -> ( ps -> th ) ) -> ( ( ( ps -> th ) -> ( ph -> ch ) ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )
20	10, 18, 19	e22 38896	. . 3 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) <-> ( ps -> th ) ) ).
21	20	in2 38830	. 2 |- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ).
22	21	in1 38787	1 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196

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)