Theorem con5VD 39136

Description: Virtual deduction proof of con5 38728. 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. con5 38728 is con5VD 39136 without virtual deductions and was automatically derived from con5VD 39136.

1::	|- (. ( ph <-> -. ps ) ->. ( ph <-> -. ps ) ).
2:1:	|- (. ( ph <-> -. ps ) ->. ( -. ps -> ph ) ).
3:2:	|- (. ( ph <-> -. ps ) ->. ( -. ph -> -. -. ps ) ).
4::	|- ( ps <-> -. -. ps )
5:3,4:	|- (. ( ph <-> -. ps ) ->. ( -. ph -> ps ) ).
qed:5:	|- ( ( ph <-> -. ps ) -> ( -. ph -> ps ) )

(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con5VD	|- ( ( ph <-> -. ps ) -> ( -. ph -> ps ) )

Proof of Theorem con5VD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 |- (. ( ph <-> -. ps ) ->. ( ph <-> -. ps ) ).
2		biimpr 210	. . . . 5 |- ( ( ph <-> -. ps ) -> ( -. ps -> ph ) )
3	1, 2	e1a 38852	. . . 4 |- (. ( ph <-> -. ps ) ->. ( -. ps -> ph ) ).
4		con3 149	. . . 4 |- ( ( -. ps -> ph ) -> ( -. ph -> -. -. ps ) )
5	3, 4	e1a 38852	. . 3 |- (. ( ph <-> -. ps ) ->. ( -. ph -> -. -. ps ) ).
6		notnotb 304	. . 3 |- ( ps <-> -. -. ps )
7		imbi2 338	. . . 4 |- ( ( ps <-> -. -. ps ) -> ( ( -. ph -> ps ) <-> ( -. ph -> -. -. ps ) ) )
8	7	biimprcd 240	. . 3 |- ( ( -. ph -> -. -. ps ) -> ( ( ps <-> -. -. ps ) -> ( -. ph -> ps ) ) )
9	5, 6, 8	e10 38919	. 2 |- (. ( ph <-> -. ps ) ->. ( -. ph -> ps ) ).
10	9	in1 38787	1 |- ( ( ph <-> -. ps ) -> ( -. ph -> ps ) )

Syntax hints:   -. wn 3   -> 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-vd1 38786

This theorem is referenced by: (None)