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Theorem imbi12VD 28965
Description: Implication form of imbi12i 316. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 28581 is imbi12VD 28965 without virtual deductions and was automatically derived from imbi12VD 28965.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ph  ->  ch ) ).
4:1,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  ch ) ).
5:2,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  th ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  ->  ( ps  ->  th ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ps  ->  th ) ).
8:1,7:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  th ) ).
9:2,8:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  ch ) ).
10:9:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ps  ->  th )  ->  ( ph  ->  ch ) ) ).
11:6,10:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ).
12:11:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi12VD  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )

Proof of Theorem imbi12VD
StepHypRef Expression
1 idn2 28690 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ch  <->  th ) ).
2 idn1 28641 . . . . . . 7  |-  (. ( ph 
<->  ps )  ->.  ( ph  <->  ps ) ).
3 idn3 28692 . . . . . . 7  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ph  ->  ch )  ->.  ( ph  ->  ch ) ).
4 bi2 189 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
54imim1d 69 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  (
( ph  ->  ch )  ->  ( ps  ->  ch ) ) )
62, 3, 5e13 28837 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  ch ) ).
7 bi1 178 . . . . . . 7  |-  ( ( ch  <->  th )  ->  ( ch  ->  th ) )
87imim2d 48 . . . . . 6  |-  ( ( ch  <->  th )  ->  (
( ps  ->  ch )  ->  ( ps  ->  th ) ) )
91, 6, 8e23 28844 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  th ) ).
109in3 28686 . . . 4  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ph  ->  ch )  -> 
( ps  ->  th )
) ).
11 idn3 28692 . . . . . . 7  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ps  ->  th )  ->.  ( ps  ->  th ) ).
12 bi1 178 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
1312imim1d 69 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  (
( ps  ->  th )  ->  ( ph  ->  th )
) )
142, 11, 13e13 28837 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  th ) ).
15 bi2 189 . . . . . . 7  |-  ( ( ch  <->  th )  ->  ( th  ->  ch ) )
1615imim2d 48 . . . . . 6  |-  ( ( ch  <->  th )  ->  (
( ph  ->  th )  ->  ( ph  ->  ch ) ) )
171, 14, 16e23 28844 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  ch ) ).
1817in3 28686 . . . 4  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ps  ->  th )  ->  ( ph  ->  ch ) ) ).
19 bi3 179 . . . 4  |-  ( ( ( ph  ->  ch )  ->  ( ps  ->  th ) )  ->  (
( ( ps  ->  th )  ->  ( ph  ->  ch ) )  -> 
( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
2010, 18, 19e22 28748 . . 3  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ).
2120in2 28682 . 2  |-  (. ( ph 
<->  ps )  ->.  ( ( ch 
<->  th )  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) ) ).
2221in1 28638 1  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
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 28637  df-vd2 28646  df-vd3 28658
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