Metamath Proof Explorer

Theorem sbc3orgVD

Description: Virtual deduction proof of the analogue of sbcor with three disjuncts. 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:: |- (. A e. B ->. A e. B ).
2:1,?: e1a |- (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
3:: |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
32:3: |- A. x ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
33:1,32,?: e10 |- (. A e. B ->. [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) ).
4:1,33,?: e11 |- (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ).
5:2,4,?: e11 |- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
6:1,?: e1a |- (. A e. B ->. ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ).
7:6,?: e1a |- (. A e. B ->. ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
8:5,7,?: e11 |- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
9:?: |- ( ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) )
10:8,9,?: e10 |- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) ).
qed:10: |- ( A e. B -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sbc3orgVD 
|- ( A e. B -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. A e. B ->. A e. B ).
2 sbcor 
 |-  ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) )
3 2 a1i 
 |-  ( A e. B -> ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) )
4 1 3 e1a 
 |-  (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
5 df-3or 
 |-  ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
6 5 bicomi 
 |-  ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
7 6 ax-gen 
 |-  A. x ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
8 spsbc 
 |-  ( A e. B -> ( A. x ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) -> [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) ) )
9 1 7 8 e10 
 |-  (. A e. B ->. [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) ).
10 sbcbig 
 |-  ( A e. B -> ( [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) <-> ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ) )
11 10 biimpd 
 |-  ( A e. B -> ( [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) -> ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ) )
12 1 9 11 e11 
 |-  (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ).
13 bitr3 
 |-  ( ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) -> ( ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ) )
14 13 com12 
 |-  ( ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) -> ( ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ) )
15 4 12 14 e11 
 |-  (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
16 sbcor 
 |-  ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )
17 16 a1i 
 |-  ( A e. B -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
18 1 17 e1a 
 |-  (. A e. B ->. ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ).
19 orbi1 
 |-  ( ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) -> ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) )
20 18 19 e1a 
 |-  (. A e. B ->. ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
21 bibi1 
 |-  ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) -> ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) <-> ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ) )
22 21 biimprd 
 |-  ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) -> ( ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ) )
23 15 20 22 e11 
 |-  (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
24 df-3or 
 |-  ( ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) )
25 24 bicomi 
 |-  ( ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) )
26 bibi1 
 |-  ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) -> ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) <-> ( ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) ) )
27 26 biimprd 
 |-  ( ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) -> ( ( ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) ) )
28 23 25 27 e10 
 |-  (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) ).
29 28 in1 
 |-  ( A e. B -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )