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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
StepHypRef 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 ) )
31, 2e1_ 31166 . . . . . 6  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ph  <->  ps ) ).
4 simp2 984 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ch  <->  th ) )
51, 4e1_ 31166 . . . . . 6  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ch  <->  th ) ).
6 pm4.39 861 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) )
76ex 434 . . . . . 6  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ) )
83, 5, 7e11 31227 . . . . 5  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
9 simp3 985 . . . . . 6  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ta  <->  et ) )
101, 9e1_ 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 ) ) )
1211ex 434 . . . . 5  |-  ( ( ( ph  \/  ch ) 
<->  ( ps  \/  th ) )  ->  (
( ta  <->  et )  ->  ( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) ) ) )
138, 10, 12e11 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 ) )
1514bicomi 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 ) ) ) )
1716com12 31 . . . 4  |-  ( ( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) )  ->  ( (
( ( ph  \/  ch )  \/  ta ) 
<->  ( ph  \/  ch  \/  ta ) )  -> 
( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ) )
1813, 15, 17e10 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 ) )
2019bicomi 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 ) ) )
2221ex 434 . . 3  |-  ( ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) )  -> 
( ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )  ->  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) ) )
2318, 20, 22e10 31233 . 2  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) ).
2423in1 31100 1  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) )
Colors of variables: wff setvar class
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)
  Copyright terms: Public domain W3C validator