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Theorem imbi13 28283
Description: Join three logical equivalences to form equivalence of implications. imbi13 28283 is imbi13VD 28650 without virtual deductions and was automatically derived from imbi13VD 28650 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 28282 . 2  |-  ( ( ch  <->  th )  ->  (
( ta  <->  et )  ->  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ) )
2 imbi12 28282 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ta )  <->  ( th  ->  et ) )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) )
31, 2syl9r 67 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 is referenced by:  trsbc  28304  trsbcVD  28653
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
  Copyright terms: Public domain W3C validator