Theorem con3ALTVD 39152

Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of [Margaris] p. 60 ( which is con3 149). The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT2 38736 is con3ALTVD 39152 without virtual deductions and was automatically derived from con3ALTVD 39152. Step i of the User's Proof corresponds to step i of the Fitch-style proof.

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

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

Assertion

Ref	Expression
con3ALTVD	|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )

Proof of Theorem con3ALTVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . 6 |- (. ( ph -> ps ) ->. ( ph -> ps ) ).
2		idn2 38838	. . . . . . 7 |- (. ( ph -> ps ) ,. -. -. ph ->. -. -. ph ).
3		notnotr 125	. . . . . . 7 |- ( -. -. ph -> ph )
4	2, 3	e2 38856	. . . . . 6 |- (. ( ph -> ps ) ,. -. -. ph ->. ph ).
5		id 22	. . . . . 6 |- ( ( ph -> ps ) -> ( ph -> ps ) )
6	1, 4, 5	e12 38951	. . . . 5 |- (. ( ph -> ps ) ,. -. -. ph ->. ps ).
7		notnot 136	. . . . 5 |- ( ps -> -. -. ps )
8	6, 7	e2 38856	. . . 4 |- (. ( ph -> ps ) ,. -. -. ph ->. -. -. ps ).
9	8	in2 38830	. . 3 |- (. ( ph -> ps ) ->. ( -. -. ph -> -. -. ps ) ).
10		con4 112	. . 3 |- ( ( -. -. ph -> -. -. ps ) -> ( -. ps -> -. ph ) )
11	9, 10	e1a 38852	. 2 |- (. ( ph -> ps ) ->. ( -. ps -> -. ph ) ).
12	11	in1 38787	1 |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4

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-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)