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Theorem imbi12VD 28649
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 28282 is imbi12VD 28649 without virtual deductions and was automatically derived from imbi12VD 28649.
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 28385 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ch  <->  th ) ).
2 idn1 28342 . . . . . . 7  |-  (. ( ph 
<->  ps )  ->.  ( ph  <->  ps ) ).
3 idn3 28387 . . . . . . 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 28523 . . . . . 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 28530 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  th ) ).
109in3 28381 . . . 4  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ph  ->  ch )  -> 
( ps  ->  th )
) ).
11 idn3 28387 . . . . . . 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 28523 . . . . . 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 28530 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  ch ) ).
1817in3 28381 . . . 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 28443 . . 3  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ).
2120in2 28377 . 2  |-  (. ( ph 
<->  ps )  ->.  ( ( ch 
<->  th )  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) ) ).
2221in1 28339 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 28338  df-vd2 28347  df-vd3 28359
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