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Theorem bj-bi3ant 31165
Description: This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
Hypothesis
Ref Expression
bj-bi3ant.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
bj-bi3ant  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch ) ) )

Proof of Theorem bj-bi3ant
StepHypRef Expression
1 biimp 196 . . 3  |-  ( ( th  <->  ta )  ->  ( th  ->  ta ) )
21imim1i 60 . 2  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( th  <->  ta )  ->  ph ) )
3 biimpr 201 . . 3  |-  ( ( th  <->  ta )  ->  ( ta  ->  th ) )
43imim1i 60 . 2  |-  ( ( ( ta  ->  th )  ->  ps )  ->  (
( th  <->  ta )  ->  ps ) )
5 bj-bi3ant.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
65imim3i 61 . 2  |-  ( ( ( th  <->  ta )  ->  ph )  ->  (
( ( th  <->  ta )  ->  ps )  ->  (
( th  <->  ta )  ->  ch ) ) )
72, 4, 6syl2im 39 1  |-  ( ( ( th  ->  ta )  ->  ph )  ->  (
( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187
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 188
This theorem is referenced by:  bj-bisym  31166
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