Theorem imbi13VD 39110

Description: Join three logical equivalences to form equivalence of implications. 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. imbi13 38726 is imbi13VD 39110 without virtual deductions and was automatically derived from imbi13VD 39110.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ta <-> et ) ).
4:2,3:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ch -> ta ) <-> ( th -> et ) ) ).
5:1,4:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ).
6:5:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ).
7:6:	|- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) ).
qed:7:	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

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

Assertion

Ref	Expression
imbi13VD	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

Proof of Theorem imbi13VD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2		idn2 38838	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3		idn3 38840	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ta <-> et ) ).
4		imbi12 336	. . . . . 6 |- ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ch -> ta ) <-> ( th -> et ) ) ) )
5	2, 3, 4	e23 38982	. . . . 5 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ch -> ta ) <-> ( th -> et ) ) ).
6		imbi12 336	. . . . 5 |- ( ( ph <-> ps ) -> ( ( ( ch -> ta ) <-> ( th -> et ) ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) )
7	1, 5, 6	e13 38975	. . . 4 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ).
8	7	in3 38834	. . 3 |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ).
9	8	in2 38830	. 2 |- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) ).
10	9	in1 38787	1 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

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)