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Theorem 3orbi123VD 29024
Description: Virtual deduction proof of 3orbi123 28656. 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_ 28790  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 28790  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 28790  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 28851  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 28851  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 28857  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 28857  |-  (. ( ( 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 28727 . . . . . . 7  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph 
<->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) ) ).
2 simp1 958 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ph  <->  ps ) )
31, 2e1_ 28790 . . . . . 6  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ph  <->  ps ) ).
4 simp2 959 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ch  <->  th ) )
51, 4e1_ 28790 . . . . . 6  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ch  <->  th ) ).
6 pm4.39 843 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) )
76ex 425 . . . . . 6  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ) )
83, 5, 7e11 28851 . . . . 5  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
9 simp3 960 . . . . . 6  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ta  <->  et ) )
101, 9e1_ 28790 . . . . 5  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ta  <->  et ) ).
11 pm4.39 843 . . . . . 6  |-  ( ( ( ( ph  \/  ch )  <->  ( ps  \/  th ) )  /\  ( ta 
<->  et ) )  -> 
( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) ) )
1211ex 425 . . . . 5  |-  ( ( ( ph  \/  ch ) 
<->  ( ps  \/  th ) )  ->  (
( ta  <->  et )  ->  ( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) ) ) )
138, 10, 12e11 28851 . . . 4  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( (
( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et )
) ).
14 df-3or 938 . . . . 5  |-  ( (
ph  \/  ch  \/  ta )  <->  ( ( ph  \/  ch )  \/  ta ) )
1514bicomi 195 . . . 4  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  (
ph  \/  ch  \/  ta ) )
16 bitr3 28655 . . . . 5  |-  ( ( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ph  \/  ch  \/  ta ) )  -> 
( ( ( (
ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) )  -> 
( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ) )
1716com12 30 . . . 4  |-  ( ( ( ( ph  \/  ch )  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) )  ->  ( (
( ( ph  \/  ch )  \/  ta ) 
<->  ( ph  \/  ch  \/  ta ) )  -> 
( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ) )
1813, 15, 17e10 28857 . . 3  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph  \/  ch  \/  ta ) 
<->  ( ( ps  \/  th )  \/  et ) ) ).
19 df-3or 938 . . . 4  |-  ( ( ps  \/  th  \/  et )  <->  ( ( ps  \/  th )  \/  et ) )
2019bicomi 195 . . 3  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
21 bitr 691 . . . 4  |-  ( ( ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) )  /\  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) ) )  ->  (
( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) )
2221ex 425 . . 3  |-  ( ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) )  -> 
( ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )  ->  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) ) )
2318, 20, 22e10 28857 . 2  |-  (. (
( ph  <->  ps )  /\  ( ch 
<->  th )  /\  ( ta 
<->  et ) )  ->.  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) ).
2423in1 28724 1  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et )
)  ->  ( ( ph  \/  ch  \/  ta ) 
<->  ( ps  \/  th  \/  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    \/ w3o 936    /\ w3a 937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-vd1 28723
  Copyright terms: Public domain W3C validator