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Theorem imbi13VD 28650
Description: Join three logical equivalences to form equivalence of implications. 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. imbi13 28283 is imbi13VD 28650 without virtual deductions and was automatically derived from imbi13VD 28650.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ta  <->  et ) ).
4:2,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ).
5:1,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ).
7:6:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) ).
qed:7:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi13VD  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) )

Proof of Theorem imbi13VD
StepHypRef Expression
1 idn1 28342 . . . . 5  |-  (. ( ph 
<->  ps )  ->.  ( ph  <->  ps ) ).
2 idn2 28385 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ch  <->  th ) ).
3 idn3 28387 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ta  <->  et ) ).
4 imbi12 28282 . . . . . 6  |-  ( ( ch  <->  th )  ->  (
( ta  <->  et )  ->  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ) )
52, 3, 4e23 28530 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ).
6 imbi12 28282 . . . . 5  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ta )  <->  ( th  ->  et ) )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) )
71, 5, 6e13 28523 . . . 4  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ).
87in3 28381 . . 3  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ta 
<->  et )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ).
98in2 28377 . 2  |-  (. ( ph 
<->  ps )  ->.  ( ( ch 
<->  th )  ->  (
( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) ).
109in1 28339 1  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) )
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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