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Theorem imbi13 36287
Description: Join three logical equivalences to form equivalence of implications. imbi13 36287 is imbi13VD 36685 without virtual deductions and was automatically derived from imbi13VD 36685 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi13  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) )

Proof of Theorem imbi13
StepHypRef Expression
1 imbi12 320 . 2  |-  ( ( ch  <->  th )  ->  (
( ta  <->  et )  ->  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ) )
2 imbi12 320 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ta )  <->  ( th  ->  et ) )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) )
31, 2syl9r 71 1  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184
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 185
This theorem is referenced by:  trsbc  36311  trsbcVD  36688
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