Theorem 3orbi123VD 31403

Description: Virtual deduction proof of 3orbi123 31032. 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 ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ).
2:1,?: e1_ 31166	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ph <-> ps ) ).
3:1,?: e1_ 31166	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ch <-> th ) ).
4:1,?: e1_ 31166	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ta <-> et ) ).
5:2,3,?: e11 31227	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch ) <-> ( ps \/ th ) ) ).
6:5,4,?: e11 31227	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
7:?:	|- ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) )
8:6,7,?: e10 31233	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
9:?:	|- ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) )
10:8,9,?: e10 31233	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) ).
qed:10:	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
3orbi123VD	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )

Proof of Theorem 3orbi123VD

Step	Hyp	Ref	Expression
1		idn1 31103	. . . . . . 7 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ).
2		simp1 983	. . . . . . 7 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ph <-> ps ) )
3	1, 2	e1_ 31166	. . . . . 6 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ph <-> ps ) ).
4		simp2 984	. . . . . . 7 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ch <-> th ) )
5	1, 4	e1_ 31166	. . . . . 6 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ch <-> th ) ).
6		pm4.39 861	. . . . . . 7 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph \/ ch ) <-> ( ps \/ th ) ) )
7	6	ex 434	. . . . . 6 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph \/ ch ) <-> ( ps \/ th ) ) ) )
8	3, 5, 7	e11 31227	. . . . 5 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch ) <-> ( ps \/ th ) ) ).
9		simp3 985	. . . . . 6 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ta <-> et ) )
10	1, 9	e1_ 31166	. . . . 5 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ta <-> et ) ).
11		pm4.39 861	. . . . . 6 |- ( ( ( ( ph \/ ch ) <-> ( ps \/ th ) ) /\ ( ta <-> et ) ) -> ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) )
12	11	ex 434	. . . . 5 |- ( ( ( ph \/ ch ) <-> ( ps \/ th ) ) -> ( ( ta <-> et ) -> ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ) )
13	8, 10, 12	e11 31227	. . . 4 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
14		df-3or 961	. . . . 5 |- ( ( ph \/ ch \/ ta ) <-> ( ( ph \/ ch ) \/ ta ) )
15	14	bicomi 202	. . . 4 |- ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) )
16		bitr3 31031	. . . . 5 |- ( ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) ) -> ( ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ) )
17	16	com12 31	. . . 4 |- ( ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) -> ( ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ) )
18	13, 15, 17	e10 31233	. . 3 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
19		df-3or 961	. . . 4 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
20	19	bicomi 202	. . 3 |- ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) )
21		bitr 703	. . . 4 |- ( ( ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) /\ ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )
22	21	ex 434	. . 3 |- ( ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) -> ( ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) ) )
23	18, 20, 22	e10 31233	. 2 |- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) ).
24	23	in1 31100	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   \/ w3o 959   /\ w3a 960

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 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-vd1 31099

This theorem is referenced by: (None)