Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3impexpVD Structured version   Unicode version

Theorem 3impexpVD 36892
Description: Virtual deduction proof of 3impexp 1227. 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 36711  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1a 36644  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1a 36644  |-  (. ( ( 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,?: e1a 36644  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1a 36644  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 36708  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 36795  |-  ( ( ( 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 36582 . . . . . 6  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps  /\ 
ch )  ->  th ) ).
2 df-3an 984 . . . . . 6  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
3 imbi1 324 . . . . . . 7  |-  ( ( ( ph  /\  ps  /\ 
ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  (
( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ( (
ph  /\  ps )  /\  ch )  ->  th )
) )
43biimpcd 227 . . . . . 6  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  ->  ( ( ( ph  /\ 
ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  ( ( (
ph  /\  ps )  /\  ch )  ->  th )
) )
51, 2, 4e10 36711 . . . . 5  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
6 pm3.3 445 . . . . 5  |-  ( ( ( ( ph  /\  ps )  /\  ch )  ->  th )  ->  (
( ph  /\  ps )  ->  ( ch  ->  th )
) )
75, 6e1a 36644 . . . 4  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
8 pm3.3 445 . . . 4  |-  ( ( ( ph  /\  ps )  ->  ( ch  ->  th ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
97, 8e1a 36644 . . 3  |-  (. (
( ph  /\  ps  /\  ch )  ->  th )  ->.  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) ).
109in1 36579 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
11 idn1 36582 . . . . . 6  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
12 pm3.31 446 . . . . . 6  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  (
( ph  /\  ps )  ->  ( ch  ->  th )
) )
1311, 12e1a 36644 . . . . 5  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ( ph  /\  ps )  -> 
( ch  ->  th )
) ).
14 pm3.31 446 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  ( ch  ->  th ) )  ->  (
( ( ph  /\  ps )  /\  ch )  ->  th ) )
1513, 14e1a 36644 . . . 4  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( (
( ph  /\  ps )  /\  ch )  ->  th ) ).
163biimprd 226 . . . 4  |-  ( ( ( ph  /\  ps  /\ 
ch )  <->  ( ( ph  /\  ps )  /\  ch ) )  ->  (
( ( ( ph  /\ 
ps )  /\  ch )  ->  th )  ->  (
( ph  /\  ps  /\  ch )  ->  th )
) )
172, 15, 16e01 36708 . . 3  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
1817in1 36579 . 2  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  (
( ph  /\  ps  /\  ch )  ->  th )
)
19 impbi 189 . 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 36795 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982
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 188  df-an 372  df-3an 984  df-vd1 36578
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator