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Theorem bi123imp0 28293
Description: Similar to 3imp 1147 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
Hypothesis
Ref Expression
bi23imp0.1  |-  ( ph  <->  ( ps  <->  ( ch  <->  th )
) )
Assertion
Ref Expression
bi123imp0  |-  ( (
ph  /\  ps  /\  ch )  ->  th )

Proof of Theorem bi123imp0
StepHypRef Expression
1 bi23imp0.1 . . 3  |-  ( ph  <->  ( ps  <->  ( ch  <->  th )
) )
2 bi1 179 . . . 4  |-  ( ( ps  <->  ( ch  <->  th )
)  ->  ( ps  ->  ( ch  <->  th )
) )
3 bi1 179 . . . 4  |-  ( ( ch  <->  th )  ->  ( ch  ->  th ) )
42, 3syl6 31 . . 3  |-  ( ( ps  <->  ( ch  <->  th )
)  ->  ( ps  ->  ( ch  ->  th )
) )
51, 4sylbi 188 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
653imp 1147 1  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936
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  df-an 361  df-3an 938
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