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Theorem imbi13 28319
Description: Join three logical equivalences to form equivalence of implications. imbi13 28319 is imbi13VD 28699 without virtual deductions and was automatically derived from imbi13VD 28699 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 28318 . 2  |-  ( ( ch  <->  th )  ->  (
( ta  <->  et )  ->  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ) )
2 imbi12 28318 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ta )  <->  ( th  ->  et ) )  ->  (
( ph  ->  ( ch 
->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) )
31, 2syl9r 69 1  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta )
)  <->  ( ps  ->  ( th  ->  et )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177
This theorem is referenced by:  trsbc  28340  trsbcVD  28702
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 178
  Copyright terms: Public domain W3C validator