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Theorem imbi13VD 28966
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 28582 is imbi13VD 28966 without virtual deductions and was automatically derived from imbi13VD 28966.
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 28641 . . . . 5  |-  (. ( ph 
<->  ps )  ->.  ( ph  <->  ps ) ).
2 idn2 28690 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ch  <->  th ) ).
3 idn3 28692 . . . . . 6  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ta  <->  et ) ).
4 imbi12 28581 . . . . . 6  |-  ( ( ch  <->  th )  ->  (
( ta  <->  et )  ->  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ) )
52, 3, 4e23 28844 . . . . 5  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ).
6 imbi12 28581 . . . . 5  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ta )  <->  ( th  ->  et ) )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) )
71, 5, 6e13 28837 . . . 4  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th ) ,. ( ta 
<->  et )  ->.  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ).
87in3 28686 . . 3  |-  (. ( ph 
<->  ps ) ,. ( ch 
<->  th )  ->.  ( ( ta 
<->  et )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ).
98in2 28682 . 2  |-  (. ( ph 
<->  ps )  ->.  ( ( ch 
<->  th )  ->  (
( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) ).
109in1 28638 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 28637  df-vd2 28646  df-vd3 28658
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