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Metamath Proof Explorer

Theorem orbi1rVD

Description: Virtual deduction proof of orbi1r . 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 ) ->. ( ch \/ ph ) ).
3:2,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
4:1,3,?: e12 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
5:4,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
6:5: |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
7:: |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
8:7,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
9:1,8,?: e12 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
10:9,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
11:10: |- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
12:6,11,?: e11 |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ).
qed:12: |- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion orbi1rVD 
|- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2 idn2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ph ) ).
3 pm1.4 
 |-  ( ( ch \/ ph ) -> ( ph \/ ch ) )
4 2 3 e2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
5 orbi1 
 |-  ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
6 5 biimpd 
 |-  ( ( ph <-> ps ) -> ( ( ph \/ ch ) -> ( ps \/ ch ) ) )
7 1 4 6 e12 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
8 pm1.4 
 |-  ( ( ps \/ ch ) -> ( ch \/ ps ) )
9 7 8 e2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
10 9 in2 
 |-  (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
11 idn2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
12 pm1.4 
 |-  ( ( ch \/ ps ) -> ( ps \/ ch ) )
13 11 12 e2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
14 5 biimprd 
 |-  ( ( ph <-> ps ) -> ( ( ps \/ ch ) -> ( ph \/ ch ) ) )
15 1 13 14 e12 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
16 pm1.4 
 |-  ( ( ph \/ ch ) -> ( ch \/ ph ) )
17 15 16 e2 
 |-  (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
18 17 in2 
 |-  (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
19 impbi 
 |-  ( ( ( ch \/ ph ) -> ( ch \/ ps ) ) -> ( ( ( ch \/ ps ) -> ( ch \/ ph ) ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ) )
20 10 18 19 e11 
 |-  (. ( ph <-> ps ) ->. ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ).
21 20 in1 
 |-  ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )