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Theorem sbc3orgVD 28138
Description: Virtual deduction proof of sbc3org 27789. 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_ 27899  |-  (. 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 27967  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 27960  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 27960  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 27899  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 27899  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 27960  |-  (. 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 27967  |-  (. 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 27836 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 sbcorg 3070 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )
) )
31, 2e1_ 27899 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( (
ph  \/  ps )  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/ 
[. A  /  x ]. ch ) ) ).
4 df-3or 935 . . . . . . . . 9  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
54bicomi 193 . . . . . . . 8  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
65ax-gen 1537 . . . . . . 7  |-  A. x
( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) )
7 spsbc 3037 . . . . . . 7  |-  ( A  e.  B  ->  ( A. x ( ( (
ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch )
)  ->  [. A  /  x ]. ( ( (
ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch )
) ) )
81, 6, 7e10 27967 . . . . . 6  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) ) ).
9 sbcbig 3071 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch ) 
<->  ( ph  \/  ps  \/  ch ) )  <->  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ) )
109biimpd 198 . . . . . 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 27960 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( (
ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
12 bitr3 27766 . . . . . 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 27 . . . . 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 27960 . . . 4  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  (
[. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
15 sbcorg 3070 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
161, 15e1_ 27899 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
17 orbi1 686 . . . . 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_ 27899 . . . 4  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/ 
[. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
19 bibi1 317 . . . . 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 214 . . . 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 27960 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
22 df-3or 935 . . . 4  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
2322bicomi 193 . . 3  |-  ( ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
24 bibi1 317 . . . 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 214 . . 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 27967 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) ).
2726in1 27833 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 176    \/ wo 357    \/ w3o 933   A.wal 1531    e. wcel 1701   [.wsbc 3025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-vd1 27832
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