Theorem orbi1rVD 39083

Description: Virtual deduction proof of orbi1r 38716. 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 38856	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
4:1,3,?: e12 38951	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
5:4,?: e2 38856	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
6:5:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
7::	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
8:7,?: e2 38856	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
9:1,8,?: e12 38951	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
10:9,?: e2 38856	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
11:10:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
12:6,11,?: e11 38913	|- (. ( 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.)

Assertion

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

Proof of Theorem orbi1rVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . 6 |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2		idn2 38838	. . . . . . 7 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ph ) ).
3		pm1.4 401	. . . . . . 7 |- ( ( ch \/ ph ) -> ( ph \/ ch ) )
4	2, 3	e2 38856	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
5		orbi1 742	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
6	5	biimpd 219	. . . . . 6 |- ( ( ph <-> ps ) -> ( ( ph \/ ch ) -> ( ps \/ ch ) ) )
7	1, 4, 6	e12 38951	. . . . 5 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
8		pm1.4 401	. . . . 5 |- ( ( ps \/ ch ) -> ( ch \/ ps ) )
9	7, 8	e2 38856	. . . 4 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
10	9	in2 38830	. . 3 |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
11		idn2 38838	. . . . . . 7 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
12		pm1.4 401	. . . . . . 7 |- ( ( ch \/ ps ) -> ( ps \/ ch ) )
13	11, 12	e2 38856	. . . . . 6 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
14	5	biimprd 238	. . . . . 6 |- ( ( ph <-> ps ) -> ( ( ps \/ ch ) -> ( ph \/ ch ) ) )
15	1, 13, 14	e12 38951	. . . . 5 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
16		pm1.4 401	. . . . 5 |- ( ( ph \/ ch ) -> ( ch \/ ph ) )
17	15, 16	e2 38856	. . . 4 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
18	17	in2 38830	. . 3 |- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
19		impbi 198	. . 3 |- ( ( ( ch \/ ph ) -> ( ch \/ ps ) ) -> ( ( ( ch \/ ps ) -> ( ch \/ ph ) ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ) )
20	10, 18, 19	e11 38913	. 2 |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ).
21	20	in1 38787	1 |- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383

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-or 385  df-an 386  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)