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Theorem bi3ant 280
Description: Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
Hypothesis
Ref Expression
bi3ant.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
bi3ant  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch ) ) )

Proof of Theorem bi3ant
StepHypRef Expression
1 bi1 178 . . 3  |-  ( ( th  <->  ta )  ->  ( th  ->  ta ) )
21imim1i 54 . 2  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( th  <->  ta )  ->  ph ) )
3 bi2 189 . . 3  |-  ( ( th  <->  ta )  ->  ( ta  ->  th ) )
43imim1i 54 . 2  |-  ( ( ( ta  ->  th )  ->  ps )  ->  (
( th  <->  ta )  ->  ps ) )
5 bi3ant.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
65imim3i 55 . 2  |-  ( ( ( th  <->  ta )  ->  ph )  ->  (
( ( th  <->  ta )  ->  ps )  ->  (
( th  <->  ta )  ->  ch ) ) )
72, 4, 6syl2im 34 1  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
This theorem is referenced by:  bisym  281
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