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Theorem biantr 897
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 309 . 2  |-  ( ( ch  <->  ps )  ->  (
( ph  <->  ch )  <->  ( ph  <->  ps ) ) )
32biimparc 473 1  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  ps )
)  ->  ( ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  bm1.1  2268  aiffbtbat  27876  bitr3VD  28625  sbcoreleleqVD  28635  trsbcVD  28653  sbcssVD  28659
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  df-an 360
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