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Theorem sbc3orgVD 29037
Description: Virtual deduction proof of sbc3org 28690. 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,?: e1_ 28802  |-  (. 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 28869  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 28863  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 28863  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 28802  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 28802  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 28863  |-  (. 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 28869  |-  (. 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.)
Assertion
Ref Expression
sbc3orgVD  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )

Proof of Theorem sbc3orgVD
StepHypRef Expression
1 idn1 28739 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 sbcorg 3208 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )
) )
31, 2e1_ 28802 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( (
ph  \/  ps )  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/ 
[. A  /  x ]. ch ) ) ).
4 df-3or 938 . . . . . . . . 9  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
54bicomi 195 . . . . . . . 8  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
65ax-gen 1556 . . . . . . 7  |-  A. x
( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) )
7 spsbc 3175 . . . . . . 7  |-  ( A  e.  B  ->  ( A. x ( ( (
ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch )
)  ->  [. A  /  x ]. ( ( (
ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch )
) ) )
81, 6, 7e10 28869 . . . . . 6  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) ) ).
9 sbcbig 3209 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) )  <->  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ) )
109biimpd 200 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) )  -> 
( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<-> 
[. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ) )
111, 8, 10e11 28863 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( (
ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
12 bitr3 28667 . . . . . 6  |-  ( (
[. 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 ) ) ) )
1312com12 30 . . . . 5  |-  ( (
[. 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 ) ) ) )
143, 11, 13e11 28863 . . . 4  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  (
[. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
15 sbcorg 3208 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
161, 15e1_ 28802 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
17 orbi1 688 . . . . 5  |-  ( (
[. 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 ) ) )
1816, 17e1_ 28802 . . . 4  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/ 
[. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
19 bibi1 319 . . . . 5  |-  ( (
[. 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 )
) ) )
2019biimprd 216 . . . 4  |-  ( (
[. 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 ) ) ) )
2114, 18, 20e11 28863 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
22 df-3or 938 . . . 4  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
2322bicomi 195 . . 3  |-  ( ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
24 bibi1 319 . . . 4  |-  ( (
[. 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 ) ) ) )
2524biimprd 216 . . 3  |-  ( (
[. 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 ) ) ) )
2621, 23, 25e10 28869 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) ).
2726in1 28736 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    \/ w3o 936   A.wal 1550    e. wcel 1726   [.wsbc 3163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-vd1 28735
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