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Theorem biantr 894
Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
biantr  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  ps )
)  ->  ( ph  <->  ch ) )

Proof of Theorem biantr
StepHypRef Expression
1 id 19 . . 3  |-  ( ( ch  <->  ps )  ->  ( ch 
<->  ps ) )
21bibi2d 230 . 2  |-  ( ( ch  <->  ps )  ->  (
( ph  <->  ch )  <->  ( ph  <->  ps ) ) )
32biimparc 293 1  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  ps )
)  ->  ( ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bm1.1  2067  bezoutlemmo  10528
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