Metamath Proof Explorer

Theorem con5VD

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

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.)

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

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. ( ph <-> -. ps ) ->. ( ph <-> -. ps ) ).
2 biimpr 
 |-  ( ( ph <-> -. ps ) -> ( -. ps -> ph ) )
3 1 2 e1a 
 |-  (. ( ph <-> -. ps ) ->. ( -. ps -> ph ) ).
4 con3 
 |-  ( ( -. ps -> ph ) -> ( -. ph -> -. -. ps ) )
5 3 4 e1a 
 |-  (. ( ph <-> -. ps ) ->. ( -. ph -> -. -. ps ) ).
6 notnotb 
 |-  ( ps <-> -. -. ps )
7 imbi2 
 |-  ( ( ps <-> -. -. ps ) -> ( ( -. ph -> ps ) <-> ( -. ph -> -. -. ps ) ) )
8 7 biimprcd 
 |-  ( ( -. ph -> -. -. ps ) -> ( ( ps <-> -. -. ps ) -> ( -. ph -> ps ) ) )
9 5 6 8 e10 
 |-  (. ( ph <-> -. ps ) ->. ( -. ph -> ps ) ).
10 9 in1 
 |-  ( ( ph <-> -. ps ) -> ( -. ph -> ps ) )