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Theorem 3impexpVD 28705
Description: Virtual deduction proof of 3impexp 1356. 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 )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
2::  |-  ( ( ph  /\  ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )
3:1,2,?: e10 28529  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 28461  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 28461  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
6:5:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
7::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
8:7,?: e1_ 28461  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 28461  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 28525  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 28614  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
3impexpVD  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) )

Proof of Theorem 3impexpVD
StepHypRef Expression
1 idn1 28398 . . . . . 6  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps  /\ 
ch )  ->  th ) ).
2 df-3an 936 . . . . . 6  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
3 imbi1 313 . . . . . . 7  |-  ( ( ( ph  /\  ps  /\ 
ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  (
( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ( (
ph  /\  ps )  /\  ch )  ->  th )
) )
43biimpcd 215 . . . . . 6  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  ->  ( ( ( ph  /\ 
ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  ( ( (
ph  /\  ps )  /\  ch )  ->  th )
) )
51, 2, 4e10 28529 . . . . 5  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
6 pm3.3 431 . . . . 5  |-  ( ( ( ( ph  /\  ps )  /\  ch )  ->  th )  ->  (
( ph  /\  ps )  ->  ( ch  ->  th )
) )
75, 6e1_ 28461 . . . 4  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
8 pm3.3 431 . . . 4  |-  ( ( ( ph  /\  ps )  ->  ( ch  ->  th ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
97, 8e1_ 28461 . . 3  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) ).
109in1 28395 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
11 idn1 28398 . . . . . 6  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
12 pm3.31 432 . . . . . 6  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  (
( ph  /\  ps )  ->  ( ch  ->  th )
) )
1311, 12e1_ 28461 . . . . 5  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ( ph  /\  ps )  -> 
( ch  ->  th )
) ).
14 pm3.31 432 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  ( ch  ->  th ) )  ->  (
( ( ph  /\  ps )  /\  ch )  ->  th ) )
1513, 14e1_ 28461 . . . 4  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( (
( ph  /\  ps )  /\  ch )  ->  th ) ).
163biimprd 214 . . . 4  |-  ( ( ( ph  /\  ps  /\ 
ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  (
( ( ( ph  /\ 
ps )  /\  ch )  ->  th )  ->  (
( ph  /\  ps  /\  ch )  ->  th )
) )
172, 15, 16e01 28525 . . 3  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
1817in1 28395 . 2  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  (
( ph  /\  ps  /\  ch )  ->  th )
)
19 bi3 179 . 2  |-  ( ( ( ( ph  /\  ps  /\  ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )  ->  ( (
( ph  ->  ( ps 
->  ( ch  ->  th )
) )  ->  (
( ph  /\  ps  /\  ch )  ->  th )
)  ->  ( (
( ph  /\  ps  /\  ch )  ->  th )  <->  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) ) ) )
2010, 18, 19e00 28614 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934
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 177  df-an 360  df-3an 936  df-vd1 28394
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