Metamath Proof Explorer

Theorem 3ornot23VD

Description: Virtual deduction proof of 3ornot23 . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1:: |- (. ( -. ph /\ -. ps ) ->. ( -. ph /\ -. ps ) ).
2:: |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ph \/ ps ) ).
3:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ph ).
4:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ps ).
5:3,4,?: e11 |- (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ).
6:2,?: e2 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ).
7:5,6,?: e12 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ch ).
8:7: |- (. ( -. ph /\ -. ps ) ->. ( ( ch \/ ph \/ ps ) -> ch ) ).
qed:8: |- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3ornot23VD 
|- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. ( -. ph /\ -. ps ) ->. ( -. ph /\ -. ps ) ).
2 simpl 
 |-  ( ( -. ph /\ -. ps ) -> -. ph )
3 1 2 e1a 
 |-  (. ( -. ph /\ -. ps ) ->. -. ph ).
4 simpr 
 |-  ( ( -. ph /\ -. ps ) -> -. ps )
5 1 4 e1a 
 |-  (. ( -. ph /\ -. ps ) ->. -. ps ).
6 ioran 
 |-  ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
7 6 simplbi2 
 |-  ( -. ph -> ( -. ps -> -. ( ph \/ ps ) ) )
8 3 5 7 e11 
 |-  (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ).
9 idn2 
 |-  (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ph \/ ps ) ).
10 3orass 
 |-  ( ( ch \/ ph \/ ps ) <-> ( ch \/ ( ph \/ ps ) ) )
11 10 biimpi 
 |-  ( ( ch \/ ph \/ ps ) -> ( ch \/ ( ph \/ ps ) ) )
12 9 11 e2 
 |-  (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ).
13 orel2 
 |-  ( -. ( ph \/ ps ) -> ( ( ch \/ ( ph \/ ps ) ) -> ch ) )
14 8 12 13 e12 
 |-  (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ch ).
15 14 in2 
 |-  (. ( -. ph /\ -. ps ) ->. ( ( ch \/ ph \/ ps ) -> ch ) ).
16 15 in1 
 |-  ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )