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Theorem 3biant1d 1373
Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 509. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
Hypothesis
Ref Expression
3biantd.1  |-  ( ph  ->  th )
Assertion
Ref Expression
3biant1d  |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ch  /\  ps ) ) )

Proof of Theorem 3biant1d
StepHypRef Expression
1 3biantd.1 . . 3  |-  ( ph  ->  th )
21biantrurd 510 . 2  |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ( ch  /\  ps )
) ) )
3 3anass 986 . 2  |-  ( ( th  /\  ch  /\  ps )  <->  ( th  /\  ( ch  /\  ps )
) )
42, 3syl6bbr 266 1  |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ch  /\  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982
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  df-an 372  df-3an 984
This theorem is referenced by:  metuel2  21504  itgsubst  22875  clwlkisclwwlk  25359  itg2addnclem2  31698
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